venv/lib/python3.10/site-packages/scipy/special/_basic.py

#
# Author:  Travis Oliphant, 2002
#

import operator
import numpy as np
import math
import warnings
from collections import defaultdict
from heapq import heapify, heappop
from numpy import (pi, asarray, floor, isscalar, iscomplex, sqrt, where, mgrid,
                   sin, place, issubdtype, extract, inexact, nan, zeros, sinc)
from . import _ufuncs
from ._ufuncs import (mathieu_a, mathieu_b, iv, jv, gamma,
                      psi, hankel1, hankel2, yv, kv, poch, binom,
                      _stirling2_inexact)
from . import _specfun
from ._comb import _comb_int
from scipy._lib.deprecation import _NoValue, _deprecate_positional_args


__all__ = [
    'ai_zeros',
    'assoc_laguerre',
    'bei_zeros',
    'beip_zeros',
    'ber_zeros',
    'bernoulli',
    'berp_zeros',
    'bi_zeros',
    'clpmn',
    'comb',
    'digamma',
    'diric',
    'erf_zeros',
    'euler',
    'factorial',
    'factorial2',
    'factorialk',
    'fresnel_zeros',
    'fresnelc_zeros',
    'fresnels_zeros',
    'h1vp',
    'h2vp',
    'ivp',
    'jn_zeros',
    'jnjnp_zeros',
    'jnp_zeros',
    'jnyn_zeros',
    'jvp',
    'kei_zeros',
    'keip_zeros',
    'kelvin_zeros',
    'ker_zeros',
    'kerp_zeros',
    'kvp',
    'lmbda',
    'lpmn',
    'lpn',
    'lqmn',
    'lqn',
    'mathieu_even_coef',
    'mathieu_odd_coef',
    'obl_cv_seq',
    'pbdn_seq',
    'pbdv_seq',
    'pbvv_seq',
    'perm',
    'polygamma',
    'pro_cv_seq',
    'riccati_jn',
    'riccati_yn',
    'sinc',
    'stirling2',
    'y0_zeros',
    'y1_zeros',
    'y1p_zeros',
    'yn_zeros',
    'ynp_zeros',
    'yvp',
    'zeta'
]


# mapping k to last n such that factorialk(n, k) < np.iinfo(np.int64).max
_FACTORIALK_LIMITS_64BITS = {1: 20, 2: 33, 3: 44, 4: 54, 5: 65,
                             6: 74, 7: 84, 8: 93, 9: 101}
# mapping k to last n such that factorialk(n, k) < np.iinfo(np.int32).max
_FACTORIALK_LIMITS_32BITS = {1: 12, 2: 19, 3: 25, 4: 31, 5: 37,
                             6: 43, 7: 47, 8: 51, 9: 56}


def _nonneg_int_or_fail(n, var_name, strict=True):
    try:
        if strict:
            # Raises an exception if float
            n = operator.index(n)
        elif n == floor(n):
            n = int(n)
        else:
            raise ValueError()
        if n < 0:
            raise ValueError()
    except (ValueError, TypeError) as err:
        raise err.__class__(f"{var_name} must be a non-negative integer") from err
    return n


def diric(x, n):
    """Periodic sinc function, also called the Dirichlet function.

    The Dirichlet function is defined as::

        diric(x, n) = sin(x * n/2) / (n * sin(x / 2)),

    where `n` is a positive integer.

    Parameters
    ----------
    x : array_like
        Input data
    n : int
        Integer defining the periodicity.

    Returns
    -------
    diric : ndarray

    Examples
    --------
    >>> import numpy as np
    >>> from scipy import special
    >>> import matplotlib.pyplot as plt

    >>> x = np.linspace(-8*np.pi, 8*np.pi, num=201)
    >>> plt.figure(figsize=(8, 8));
    >>> for idx, n in enumerate([2, 3, 4, 9]):
    ...     plt.subplot(2, 2, idx+1)
    ...     plt.plot(x, special.diric(x, n))
    ...     plt.title('diric, n={}'.format(n))
    >>> plt.show()

    The following example demonstrates that `diric` gives the magnitudes
    (modulo the sign and scaling) of the Fourier coefficients of a
    rectangular pulse.

    Suppress output of values that are effectively 0:

    >>> np.set_printoptions(suppress=True)

    Create a signal `x` of length `m` with `k` ones:

    >>> m = 8
    >>> k = 3
    >>> x = np.zeros(m)
    >>> x[:k] = 1

    Use the FFT to compute the Fourier transform of `x`, and
    inspect the magnitudes of the coefficients:

    >>> np.abs(np.fft.fft(x))
    array([ 3.        ,  2.41421356,  1.        ,  0.41421356,  1.        ,
            0.41421356,  1.        ,  2.41421356])

    Now find the same values (up to sign) using `diric`. We multiply
    by `k` to account for the different scaling conventions of
    `numpy.fft.fft` and `diric`:

    >>> theta = np.linspace(0, 2*np.pi, m, endpoint=False)
    >>> k * special.diric(theta, k)
    array([ 3.        ,  2.41421356,  1.        , -0.41421356, -1.        ,
           -0.41421356,  1.        ,  2.41421356])
    """
    x, n = asarray(x), asarray(n)
    n = asarray(n + (x-x))
    x = asarray(x + (n-n))
    if issubdtype(x.dtype, inexact):
        ytype = x.dtype
    else:
        ytype = float
    y = zeros(x.shape, ytype)

    # empirical minval for 32, 64 or 128 bit float computations
    # where sin(x/2) < minval, result is fixed at +1 or -1
    if np.finfo(ytype).eps < 1e-18:
        minval = 1e-11
    elif np.finfo(ytype).eps < 1e-15:
        minval = 1e-7
    else:
        minval = 1e-3

    mask1 = (n <= 0) | (n != floor(n))
    place(y, mask1, nan)

    x = x / 2
    denom = sin(x)
    mask2 = (1-mask1) & (abs(denom) < minval)
    xsub = extract(mask2, x)
    nsub = extract(mask2, n)
    zsub = xsub / pi
    place(y, mask2, pow(-1, np.round(zsub)*(nsub-1)))

    mask = (1-mask1) & (1-mask2)
    xsub = extract(mask, x)
    nsub = extract(mask, n)
    dsub = extract(mask, denom)
    place(y, mask, sin(nsub*xsub)/(nsub*dsub))
    return y


def jnjnp_zeros(nt):
    """Compute zeros of integer-order Bessel functions Jn and Jn'.

    Results are arranged in order of the magnitudes of the zeros.

    Parameters
    ----------
    nt : int
        Number (<=1200) of zeros to compute

    Returns
    -------
    zo[l-1] : ndarray
        Value of the lth zero of Jn(x) and Jn'(x). Of length `nt`.
    n[l-1] : ndarray
        Order of the Jn(x) or Jn'(x) associated with lth zero. Of length `nt`.
    m[l-1] : ndarray
        Serial number of the zeros of Jn(x) or Jn'(x) associated
        with lth zero. Of length `nt`.
    t[l-1] : ndarray
        0 if lth zero in zo is zero of Jn(x), 1 if it is a zero of Jn'(x). Of
        length `nt`.

    See Also
    --------
    jn_zeros, jnp_zeros : to get separated arrays of zeros.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt > 1200):
        raise ValueError("Number must be integer <= 1200.")
    nt = int(nt)
    n, m, t, zo = _specfun.jdzo(nt)
    return zo[1:nt+1], n[:nt], m[:nt], t[:nt]


def jnyn_zeros(n, nt):
    """Compute nt zeros of Bessel functions Jn(x), Jn'(x), Yn(x), and Yn'(x).

    Returns 4 arrays of length `nt`, corresponding to the first `nt`
    zeros of Jn(x), Jn'(x), Yn(x), and Yn'(x), respectively. The zeros
    are returned in ascending order.

    Parameters
    ----------
    n : int
        Order of the Bessel functions
    nt : int
        Number (<=1200) of zeros to compute

    Returns
    -------
    Jn : ndarray
        First `nt` zeros of Jn
    Jnp : ndarray
        First `nt` zeros of Jn'
    Yn : ndarray
        First `nt` zeros of Yn
    Ynp : ndarray
        First `nt` zeros of Yn'

    See Also
    --------
    jn_zeros, jnp_zeros, yn_zeros, ynp_zeros

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    Compute the first three roots of :math:`J_1`, :math:`J_1'`,
    :math:`Y_1` and :math:`Y_1'`.

    >>> from scipy.special import jnyn_zeros
    >>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3)
    >>> jn_roots, yn_roots
    (array([ 3.83170597,  7.01558667, 10.17346814]),
     array([2.19714133, 5.42968104, 8.59600587]))

    Plot :math:`J_1`, :math:`J_1'`, :math:`Y_1`, :math:`Y_1'` and their roots.

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy.special import jnyn_zeros, jvp, jn, yvp, yn
    >>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3)
    >>> fig, ax = plt.subplots()
    >>> xmax= 11
    >>> x = np.linspace(0, xmax)
    >>> x[0] += 1e-15
    >>> ax.plot(x, jn(1, x), label=r"$J_1$", c='r')
    >>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$", c='b')
    >>> ax.plot(x, yn(1, x), label=r"$Y_1$", c='y')
    >>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$", c='c')
    >>> zeros = np.zeros((3, ))
    >>> ax.scatter(jn_roots, zeros, s=30, c='r', zorder=5,
    ...            label=r"$J_1$ roots")
    >>> ax.scatter(jnp_roots, zeros, s=30, c='b', zorder=5,
    ...            label=r"$J_1'$ roots")
    >>> ax.scatter(yn_roots, zeros, s=30, c='y', zorder=5,
    ...            label=r"$Y_1$ roots")
    >>> ax.scatter(ynp_roots, zeros, s=30, c='c', zorder=5,
    ...            label=r"$Y_1'$ roots")
    >>> ax.hlines(0, 0, xmax, color='k')
    >>> ax.set_ylim(-0.6, 0.6)
    >>> ax.set_xlim(0, xmax)
    >>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75))
    >>> plt.tight_layout()
    >>> plt.show()
    """
    if not (isscalar(nt) and isscalar(n)):
        raise ValueError("Arguments must be scalars.")
    if (floor(n) != n) or (floor(nt) != nt):
        raise ValueError("Arguments must be integers.")
    if (nt <= 0):
        raise ValueError("nt > 0")
    return _specfun.jyzo(abs(n), nt)


def jn_zeros(n, nt):
    r"""Compute zeros of integer-order Bessel functions Jn.

    Compute `nt` zeros of the Bessel functions :math:`J_n(x)` on the
    interval :math:`(0, \infty)`. The zeros are returned in ascending
    order. Note that this interval excludes the zero at :math:`x = 0`
    that exists for :math:`n > 0`.

    Parameters
    ----------
    n : int
        Order of Bessel function
    nt : int
        Number of zeros to return

    Returns
    -------
    ndarray
        First `nt` zeros of the Bessel function.

    See Also
    --------
    jv: Real-order Bessel functions of the first kind
    jnp_zeros: Zeros of :math:`Jn'`

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    Compute the first four positive roots of :math:`J_3`.

    >>> from scipy.special import jn_zeros
    >>> jn_zeros(3, 4)
    array([ 6.3801619 ,  9.76102313, 13.01520072, 16.22346616])

    Plot :math:`J_3` and its first four positive roots. Note
    that the root located at 0 is not returned by `jn_zeros`.

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy.special import jn, jn_zeros
    >>> j3_roots = jn_zeros(3, 4)
    >>> xmax = 18
    >>> xmin = -1
    >>> x = np.linspace(xmin, xmax, 500)
    >>> fig, ax = plt.subplots()
    >>> ax.plot(x, jn(3, x), label=r'$J_3$')
    >>> ax.scatter(j3_roots, np.zeros((4, )), s=30, c='r',
    ...            label=r"$J_3$_Zeros", zorder=5)
    >>> ax.scatter(0, 0, s=30, c='k',
    ...            label=r"Root at 0", zorder=5)
    >>> ax.hlines(0, 0, xmax, color='k')
    >>> ax.set_xlim(xmin, xmax)
    >>> plt.legend()
    >>> plt.show()
    """
    return jnyn_zeros(n, nt)[0]


def jnp_zeros(n, nt):
    r"""Compute zeros of integer-order Bessel function derivatives Jn'.

    Compute `nt` zeros of the functions :math:`J_n'(x)` on the
    interval :math:`(0, \infty)`. The zeros are returned in ascending
    order. Note that this interval excludes the zero at :math:`x = 0`
    that exists for :math:`n > 1`.

    Parameters
    ----------
    n : int
        Order of Bessel function
    nt : int
        Number of zeros to return

    Returns
    -------
    ndarray
        First `nt` zeros of the Bessel function.

    See Also
    --------
    jvp: Derivatives of integer-order Bessel functions of the first kind
    jv: Float-order Bessel functions of the first kind

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    Compute the first four roots of :math:`J_2'`.

    >>> from scipy.special import jnp_zeros
    >>> jnp_zeros(2, 4)
    array([ 3.05423693,  6.70613319,  9.96946782, 13.17037086])

    As `jnp_zeros` yields the roots of :math:`J_n'`, it can be used to
    compute the locations of the peaks of :math:`J_n`. Plot
    :math:`J_2`, :math:`J_2'` and the locations of the roots of :math:`J_2'`.

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy.special import jn, jnp_zeros, jvp
    >>> j2_roots = jnp_zeros(2, 4)
    >>> xmax = 15
    >>> x = np.linspace(0, xmax, 500)
    >>> fig, ax = plt.subplots()
    >>> ax.plot(x, jn(2, x), label=r'$J_2$')
    >>> ax.plot(x, jvp(2, x, 1), label=r"$J_2'$")
    >>> ax.hlines(0, 0, xmax, color='k')
    >>> ax.scatter(j2_roots, np.zeros((4, )), s=30, c='r',
    ...            label=r"Roots of $J_2'$", zorder=5)
    >>> ax.set_ylim(-0.4, 0.8)
    >>> ax.set_xlim(0, xmax)
    >>> plt.legend()
    >>> plt.show()
    """
    return jnyn_zeros(n, nt)[1]


def yn_zeros(n, nt):
    r"""Compute zeros of integer-order Bessel function Yn(x).

    Compute `nt` zeros of the functions :math:`Y_n(x)` on the interval
    :math:`(0, \infty)`. The zeros are returned in ascending order.

    Parameters
    ----------
    n : int
        Order of Bessel function
    nt : int
        Number of zeros to return

    Returns
    -------
    ndarray
        First `nt` zeros of the Bessel function.

    See Also
    --------
    yn: Bessel function of the second kind for integer order
    yv: Bessel function of the second kind for real order

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    Compute the first four roots of :math:`Y_2`.

    >>> from scipy.special import yn_zeros
    >>> yn_zeros(2, 4)
    array([ 3.38424177,  6.79380751, 10.02347798, 13.20998671])

    Plot :math:`Y_2` and its first four roots.

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy.special import yn, yn_zeros
    >>> xmin = 2
    >>> xmax = 15
    >>> x = np.linspace(xmin, xmax, 500)
    >>> fig, ax = plt.subplots()
    >>> ax.hlines(0, xmin, xmax, color='k')
    >>> ax.plot(x, yn(2, x), label=r'$Y_2$')
    >>> ax.scatter(yn_zeros(2, 4), np.zeros((4, )), s=30, c='r',
    ...            label='Roots', zorder=5)
    >>> ax.set_ylim(-0.4, 0.4)
    >>> ax.set_xlim(xmin, xmax)
    >>> plt.legend()
    >>> plt.show()
    """
    return jnyn_zeros(n, nt)[2]


def ynp_zeros(n, nt):
    r"""Compute zeros of integer-order Bessel function derivatives Yn'(x).

    Compute `nt` zeros of the functions :math:`Y_n'(x)` on the
    interval :math:`(0, \infty)`. The zeros are returned in ascending
    order.

    Parameters
    ----------
    n : int
        Order of Bessel function
    nt : int
        Number of zeros to return

    Returns
    -------
    ndarray
        First `nt` zeros of the Bessel derivative function.


    See Also
    --------
    yvp

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    Compute the first four roots of the first derivative of the
    Bessel function of second kind for order 0 :math:`Y_0'`.

    >>> from scipy.special import ynp_zeros
    >>> ynp_zeros(0, 4)
    array([ 2.19714133,  5.42968104,  8.59600587, 11.74915483])

    Plot :math:`Y_0`, :math:`Y_0'` and confirm visually that the roots of
    :math:`Y_0'` are located at local extrema of :math:`Y_0`.

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy.special import yn, ynp_zeros, yvp
    >>> zeros = ynp_zeros(0, 4)
    >>> xmax = 13
    >>> x = np.linspace(0, xmax, 500)
    >>> fig, ax = plt.subplots()
    >>> ax.plot(x, yn(0, x), label=r'$Y_0$')
    >>> ax.plot(x, yvp(0, x, 1), label=r"$Y_0'$")
    >>> ax.scatter(zeros, np.zeros((4, )), s=30, c='r',
    ...            label=r"Roots of $Y_0'$", zorder=5)
    >>> for root in zeros:
    ...     y0_extremum =  yn(0, root)
    ...     lower = min(0, y0_extremum)
    ...     upper = max(0, y0_extremum)
    ...     ax.vlines(root, lower, upper, color='r')
    >>> ax.hlines(0, 0, xmax, color='k')
    >>> ax.set_ylim(-0.6, 0.6)
    >>> ax.set_xlim(0, xmax)
    >>> plt.legend()
    >>> plt.show()
    """
    return jnyn_zeros(n, nt)[3]


def y0_zeros(nt, complex=False):
    """Compute nt zeros of Bessel function Y0(z), and derivative at each zero.

    The derivatives are given by Y0'(z0) = -Y1(z0) at each zero z0.

    Parameters
    ----------
    nt : int
        Number of zeros to return
    complex : bool, default False
        Set to False to return only the real zeros; set to True to return only
        the complex zeros with negative real part and positive imaginary part.
        Note that the complex conjugates of the latter are also zeros of the
        function, but are not returned by this routine.

    Returns
    -------
    z0n : ndarray
        Location of nth zero of Y0(z)
    y0pz0n : ndarray
        Value of derivative Y0'(z0) for nth zero

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    Compute the first 4 real roots and the derivatives at the roots of
    :math:`Y_0`:

    >>> import numpy as np
    >>> from scipy.special import y0_zeros
    >>> zeros, grads = y0_zeros(4)
    >>> with np.printoptions(precision=5):
    ...     print(f"Roots: {zeros}")
    ...     print(f"Gradients: {grads}")
    Roots: [ 0.89358+0.j  3.95768+0.j  7.08605+0.j 10.22235+0.j]
    Gradients: [-0.87942+0.j  0.40254+0.j -0.3001 +0.j  0.2497 +0.j]

    Plot the real part of :math:`Y_0` and the first four computed roots.

    >>> import matplotlib.pyplot as plt
    >>> from scipy.special import y0
    >>> xmin = 0
    >>> xmax = 11
    >>> x = np.linspace(xmin, xmax, 500)
    >>> fig, ax = plt.subplots()
    >>> ax.hlines(0, xmin, xmax, color='k')
    >>> ax.plot(x, y0(x), label=r'$Y_0$')
    >>> zeros, grads = y0_zeros(4)
    >>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r',
    ...            label=r'$Y_0$_zeros', zorder=5)
    >>> ax.set_ylim(-0.5, 0.6)
    >>> ax.set_xlim(xmin, xmax)
    >>> plt.legend(ncol=2)
    >>> plt.show()

    Compute the first 4 complex roots and the derivatives at the roots of
    :math:`Y_0` by setting ``complex=True``:

    >>> y0_zeros(4, True)
    (array([ -2.40301663+0.53988231j,  -5.5198767 +0.54718001j,
             -8.6536724 +0.54841207j, -11.79151203+0.54881912j]),
     array([ 0.10074769-0.88196771j, -0.02924642+0.5871695j ,
             0.01490806-0.46945875j, -0.00937368+0.40230454j]))
    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("Arguments must be scalar positive integer.")
    kf = 0
    kc = not complex
    return _specfun.cyzo(nt, kf, kc)


def y1_zeros(nt, complex=False):
    """Compute nt zeros of Bessel function Y1(z), and derivative at each zero.

    The derivatives are given by Y1'(z1) = Y0(z1) at each zero z1.

    Parameters
    ----------
    nt : int
        Number of zeros to return
    complex : bool, default False
        Set to False to return only the real zeros; set to True to return only
        the complex zeros with negative real part and positive imaginary part.
        Note that the complex conjugates of the latter are also zeros of the
        function, but are not returned by this routine.

    Returns
    -------
    z1n : ndarray
        Location of nth zero of Y1(z)
    y1pz1n : ndarray
        Value of derivative Y1'(z1) for nth zero

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    Compute the first 4 real roots and the derivatives at the roots of
    :math:`Y_1`:

    >>> import numpy as np
    >>> from scipy.special import y1_zeros
    >>> zeros, grads = y1_zeros(4)
    >>> with np.printoptions(precision=5):
    ...     print(f"Roots: {zeros}")
    ...     print(f"Gradients: {grads}")
    Roots: [ 2.19714+0.j  5.42968+0.j  8.59601+0.j 11.74915+0.j]
    Gradients: [ 0.52079+0.j -0.34032+0.j  0.27146+0.j -0.23246+0.j]

    Extract the real parts:

    >>> realzeros = zeros.real
    >>> realzeros
    array([ 2.19714133,  5.42968104,  8.59600587, 11.74915483])

    Plot :math:`Y_1` and the first four computed roots.

    >>> import matplotlib.pyplot as plt
    >>> from scipy.special import y1
    >>> xmin = 0
    >>> xmax = 13
    >>> x = np.linspace(xmin, xmax, 500)
    >>> zeros, grads = y1_zeros(4)
    >>> fig, ax = plt.subplots()
    >>> ax.hlines(0, xmin, xmax, color='k')
    >>> ax.plot(x, y1(x), label=r'$Y_1$')
    >>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r',
    ...            label=r'$Y_1$_zeros', zorder=5)
    >>> ax.set_ylim(-0.5, 0.5)
    >>> ax.set_xlim(xmin, xmax)
    >>> plt.legend()
    >>> plt.show()

    Compute the first 4 complex roots and the derivatives at the roots of
    :math:`Y_1` by setting ``complex=True``:

    >>> y1_zeros(4, True)
    (array([ -0.50274327+0.78624371j,  -3.83353519+0.56235654j,
             -7.01590368+0.55339305j, -10.17357383+0.55127339j]),
     array([-0.45952768+1.31710194j,  0.04830191-0.69251288j,
            -0.02012695+0.51864253j,  0.011614  -0.43203296j]))
    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("Arguments must be scalar positive integer.")
    kf = 1
    kc = not complex
    return _specfun.cyzo(nt, kf, kc)


def y1p_zeros(nt, complex=False):
    """Compute nt zeros of Bessel derivative Y1'(z), and value at each zero.

    The values are given by Y1(z1) at each z1 where Y1'(z1)=0.

    Parameters
    ----------
    nt : int
        Number of zeros to return
    complex : bool, default False
        Set to False to return only the real zeros; set to True to return only
        the complex zeros with negative real part and positive imaginary part.
        Note that the complex conjugates of the latter are also zeros of the
        function, but are not returned by this routine.

    Returns
    -------
    z1pn : ndarray
        Location of nth zero of Y1'(z)
    y1z1pn : ndarray
        Value of derivative Y1(z1) for nth zero

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    Compute the first four roots of :math:`Y_1'` and the values of
    :math:`Y_1` at these roots.

    >>> import numpy as np
    >>> from scipy.special import y1p_zeros
    >>> y1grad_roots, y1_values = y1p_zeros(4)
    >>> with np.printoptions(precision=5):
    ...     print(f"Y1' Roots: {y1grad_roots.real}")
    ...     print(f"Y1 values: {y1_values.real}")
    Y1' Roots: [ 3.68302  6.9415  10.1234  13.28576]
    Y1 values: [ 0.41673 -0.30317  0.25091 -0.21897]

    `y1p_zeros` can be used to calculate the extremal points of :math:`Y_1`
    directly. Here we plot :math:`Y_1` and the first four extrema.

    >>> import matplotlib.pyplot as plt
    >>> from scipy.special import y1, yvp
    >>> y1_roots, y1_values_at_roots = y1p_zeros(4)
    >>> real_roots = y1_roots.real
    >>> xmax = 15
    >>> x = np.linspace(0, xmax, 500)
    >>> x[0] += 1e-15
    >>> fig, ax = plt.subplots()
    >>> ax.plot(x, y1(x), label=r'$Y_1$')
    >>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$")
    >>> ax.scatter(real_roots, np.zeros((4, )), s=30, c='r',
    ...            label=r"Roots of $Y_1'$", zorder=5)
    >>> ax.scatter(real_roots, y1_values_at_roots.real, s=30, c='k',
    ...            label=r"Extrema of $Y_1$", zorder=5)
    >>> ax.hlines(0, 0, xmax, color='k')
    >>> ax.set_ylim(-0.5, 0.5)
    >>> ax.set_xlim(0, xmax)
    >>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75))
    >>> plt.tight_layout()
    >>> plt.show()
    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("Arguments must be scalar positive integer.")
    kf = 2
    kc = not complex
    return _specfun.cyzo(nt, kf, kc)


def _bessel_diff_formula(v, z, n, L, phase):
    # from AMS55.
    # L(v, z) = J(v, z), Y(v, z), H1(v, z), H2(v, z), phase = -1
    # L(v, z) = I(v, z) or exp(v*pi*i)K(v, z), phase = 1
    # For K, you can pull out the exp((v-k)*pi*i) into the caller
    v = asarray(v)
    p = 1.0
    s = L(v-n, z)
    for i in range(1, n+1):
        p = phase * (p * (n-i+1)) / i   # = choose(k, i)
        s += p*L(v-n + i*2, z)
    return s / (2.**n)


def jvp(v, z, n=1):
    """Compute derivatives of Bessel functions of the first kind.

    Compute the nth derivative of the Bessel function `Jv` with
    respect to `z`.

    Parameters
    ----------
    v : array_like or float
        Order of Bessel function
    z : complex
        Argument at which to evaluate the derivative; can be real or
        complex.
    n : int, default 1
        Order of derivative. For 0 returns the Bessel function `jv` itself.

    Returns
    -------
    scalar or ndarray
        Values of the derivative of the Bessel function.

    Notes
    -----
    The derivative is computed using the relation DLFM 10.6.7 [2]_.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    .. [2] NIST Digital Library of Mathematical Functions.
           https://dlmf.nist.gov/10.6.E7

    Examples
    --------

    Compute the Bessel function of the first kind of order 0 and
    its first two derivatives at 1.

    >>> from scipy.special import jvp
    >>> jvp(0, 1, 0), jvp(0, 1, 1), jvp(0, 1, 2)
    (0.7651976865579666, -0.44005058574493355, -0.3251471008130331)

    Compute the first derivative of the Bessel function of the first
    kind for several orders at 1 by providing an array for `v`.

    >>> jvp([0, 1, 2], 1, 1)
    array([-0.44005059,  0.3251471 ,  0.21024362])

    Compute the first derivative of the Bessel function of the first
    kind of order 0 at several points by providing an array for `z`.

    >>> import numpy as np
    >>> points = np.array([0., 1.5, 3.])
    >>> jvp(0, points, 1)
    array([-0.        , -0.55793651, -0.33905896])

    Plot the Bessel function of the first kind of order 1 and its
    first three derivatives.

    >>> import matplotlib.pyplot as plt
    >>> x = np.linspace(-10, 10, 1000)
    >>> fig, ax = plt.subplots()
    >>> ax.plot(x, jvp(1, x, 0), label=r"$J_1$")
    >>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$")
    >>> ax.plot(x, jvp(1, x, 2), label=r"$J_1''$")
    >>> ax.plot(x, jvp(1, x, 3), label=r"$J_1'''$")
    >>> plt.legend()
    >>> plt.show()
    """
    n = _nonneg_int_or_fail(n, 'n')
    if n == 0:
        return jv(v, z)
    else:
        return _bessel_diff_formula(v, z, n, jv, -1)


def yvp(v, z, n=1):
    """Compute derivatives of Bessel functions of the second kind.

    Compute the nth derivative of the Bessel function `Yv` with
    respect to `z`.

    Parameters
    ----------
    v : array_like of float
        Order of Bessel function
    z : complex
        Argument at which to evaluate the derivative
    n : int, default 1
        Order of derivative. For 0 returns the BEssel function `yv`

    Returns
    -------
    scalar or ndarray
        nth derivative of the Bessel function.

    See Also
    --------
    yv : Bessel functions of the second kind

    Notes
    -----
    The derivative is computed using the relation DLFM 10.6.7 [2]_.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    .. [2] NIST Digital Library of Mathematical Functions.
           https://dlmf.nist.gov/10.6.E7

    Examples
    --------
    Compute the Bessel function of the second kind of order 0 and
    its first two derivatives at 1.

    >>> from scipy.special import yvp
    >>> yvp(0, 1, 0), yvp(0, 1, 1), yvp(0, 1, 2)
    (0.088256964215677, 0.7812128213002889, -0.8694697855159659)

    Compute the first derivative of the Bessel function of the second
    kind for several orders at 1 by providing an array for `v`.

    >>> yvp([0, 1, 2], 1, 1)
    array([0.78121282, 0.86946979, 2.52015239])

    Compute the first derivative of the Bessel function of the
    second kind of order 0 at several points by providing an array for `z`.

    >>> import numpy as np
    >>> points = np.array([0.5, 1.5, 3.])
    >>> yvp(0, points, 1)
    array([ 1.47147239,  0.41230863, -0.32467442])

    Plot the Bessel function of the second kind of order 1 and its
    first three derivatives.

    >>> import matplotlib.pyplot as plt
    >>> x = np.linspace(0, 5, 1000)
    >>> x[0] += 1e-15
    >>> fig, ax = plt.subplots()
    >>> ax.plot(x, yvp(1, x, 0), label=r"$Y_1$")
    >>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$")
    >>> ax.plot(x, yvp(1, x, 2), label=r"$Y_1''$")
    >>> ax.plot(x, yvp(1, x, 3), label=r"$Y_1'''$")
    >>> ax.set_ylim(-10, 10)
    >>> plt.legend()
    >>> plt.show()
    """
    n = _nonneg_int_or_fail(n, 'n')
    if n == 0:
        return yv(v, z)
    else:
        return _bessel_diff_formula(v, z, n, yv, -1)


def kvp(v, z, n=1):
    """Compute derivatives of real-order modified Bessel function Kv(z)

    Kv(z) is the modified Bessel function of the second kind.
    Derivative is calculated with respect to `z`.

    Parameters
    ----------
    v : array_like of float
        Order of Bessel function
    z : array_like of complex
        Argument at which to evaluate the derivative
    n : int, default 1
        Order of derivative. For 0 returns the Bessel function `kv` itself.

    Returns
    -------
    out : ndarray
        The results

    See Also
    --------
    kv

    Notes
    -----
    The derivative is computed using the relation DLFM 10.29.5 [2]_.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 6.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    .. [2] NIST Digital Library of Mathematical Functions.
           https://dlmf.nist.gov/10.29.E5

    Examples
    --------
    Compute the modified bessel function of the second kind of order 0 and
    its first two derivatives at 1.

    >>> from scipy.special import kvp
    >>> kvp(0, 1, 0), kvp(0, 1, 1), kvp(0, 1, 2)
    (0.42102443824070834, -0.6019072301972346, 1.0229316684379428)

    Compute the first derivative of the modified Bessel function of the second
    kind for several orders at 1 by providing an array for `v`.

    >>> kvp([0, 1, 2], 1, 1)
    array([-0.60190723, -1.02293167, -3.85158503])

    Compute the first derivative of the modified Bessel function of the
    second kind of order 0 at several points by providing an array for `z`.

    >>> import numpy as np
    >>> points = np.array([0.5, 1.5, 3.])
    >>> kvp(0, points, 1)
    array([-1.65644112, -0.2773878 , -0.04015643])

    Plot the modified bessel function of the second kind and its
    first three derivatives.

    >>> import matplotlib.pyplot as plt
    >>> x = np.linspace(0, 5, 1000)
    >>> fig, ax = plt.subplots()
    >>> ax.plot(x, kvp(1, x, 0), label=r"$K_1$")
    >>> ax.plot(x, kvp(1, x, 1), label=r"$K_1'$")
    >>> ax.plot(x, kvp(1, x, 2), label=r"$K_1''$")
    >>> ax.plot(x, kvp(1, x, 3), label=r"$K_1'''$")
    >>> ax.set_ylim(-2.5, 2.5)
    >>> plt.legend()
    >>> plt.show()
    """
    n = _nonneg_int_or_fail(n, 'n')
    if n == 0:
        return kv(v, z)
    else:
        return (-1)**n * _bessel_diff_formula(v, z, n, kv, 1)


def ivp(v, z, n=1):
    """Compute derivatives of modified Bessel functions of the first kind.

    Compute the nth derivative of the modified Bessel function `Iv`
    with respect to `z`.

    Parameters
    ----------
    v : array_like or float
        Order of Bessel function
    z : array_like
        Argument at which to evaluate the derivative; can be real or
        complex.
    n : int, default 1
        Order of derivative. For 0, returns the Bessel function `iv` itself.

    Returns
    -------
    scalar or ndarray
        nth derivative of the modified Bessel function.

    See Also
    --------
    iv

    Notes
    -----
    The derivative is computed using the relation DLFM 10.29.5 [2]_.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 6.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    .. [2] NIST Digital Library of Mathematical Functions.
           https://dlmf.nist.gov/10.29.E5

    Examples
    --------
    Compute the modified Bessel function of the first kind of order 0 and
    its first two derivatives at 1.

    >>> from scipy.special import ivp
    >>> ivp(0, 1, 0), ivp(0, 1, 1), ivp(0, 1, 2)
    (1.2660658777520084, 0.565159103992485, 0.7009067737595233)

    Compute the first derivative of the modified Bessel function of the first
    kind for several orders at 1 by providing an array for `v`.

    >>> ivp([0, 1, 2], 1, 1)
    array([0.5651591 , 0.70090677, 0.29366376])

    Compute the first derivative of the modified Bessel function of the
    first kind of order 0 at several points by providing an array for `z`.

    >>> import numpy as np
    >>> points = np.array([0., 1.5, 3.])
    >>> ivp(0, points, 1)
    array([0.        , 0.98166643, 3.95337022])

    Plot the modified Bessel function of the first kind of order 1 and its
    first three derivatives.

    >>> import matplotlib.pyplot as plt
    >>> x = np.linspace(-5, 5, 1000)
    >>> fig, ax = plt.subplots()
    >>> ax.plot(x, ivp(1, x, 0), label=r"$I_1$")
    >>> ax.plot(x, ivp(1, x, 1), label=r"$I_1'$")
    >>> ax.plot(x, ivp(1, x, 2), label=r"$I_1''$")
    >>> ax.plot(x, ivp(1, x, 3), label=r"$I_1'''$")
    >>> plt.legend()
    >>> plt.show()
    """
    n = _nonneg_int_or_fail(n, 'n')
    if n == 0:
        return iv(v, z)
    else:
        return _bessel_diff_formula(v, z, n, iv, 1)


def h1vp(v, z, n=1):
    """Compute derivatives of Hankel function H1v(z) with respect to `z`.

    Parameters
    ----------
    v : array_like
        Order of Hankel function
    z : array_like
        Argument at which to evaluate the derivative. Can be real or
        complex.
    n : int, default 1
        Order of derivative. For 0 returns the Hankel function `h1v` itself.

    Returns
    -------
    scalar or ndarray
        Values of the derivative of the Hankel function.

    See Also
    --------
    hankel1

    Notes
    -----
    The derivative is computed using the relation DLFM 10.6.7 [2]_.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    .. [2] NIST Digital Library of Mathematical Functions.
           https://dlmf.nist.gov/10.6.E7

    Examples
    --------
    Compute the Hankel function of the first kind of order 0 and
    its first two derivatives at 1.

    >>> from scipy.special import h1vp
    >>> h1vp(0, 1, 0), h1vp(0, 1, 1), h1vp(0, 1, 2)
    ((0.7651976865579664+0.088256964215677j),
     (-0.44005058574493355+0.7812128213002889j),
     (-0.3251471008130329-0.8694697855159659j))

    Compute the first derivative of the Hankel function of the first kind
    for several orders at 1 by providing an array for `v`.

    >>> h1vp([0, 1, 2], 1, 1)
    array([-0.44005059+0.78121282j,  0.3251471 +0.86946979j,
           0.21024362+2.52015239j])

    Compute the first derivative of the Hankel function of the first kind
    of order 0 at several points by providing an array for `z`.

    >>> import numpy as np
    >>> points = np.array([0.5, 1.5, 3.])
    >>> h1vp(0, points, 1)
    array([-0.24226846+1.47147239j, -0.55793651+0.41230863j,
           -0.33905896-0.32467442j])
    """
    n = _nonneg_int_or_fail(n, 'n')
    if n == 0:
        return hankel1(v, z)
    else:
        return _bessel_diff_formula(v, z, n, hankel1, -1)


def h2vp(v, z, n=1):
    """Compute derivatives of Hankel function H2v(z) with respect to `z`.

    Parameters
    ----------
    v : array_like
        Order of Hankel function
    z : array_like
        Argument at which to evaluate the derivative. Can be real or
        complex.
    n : int, default 1
        Order of derivative. For 0 returns the Hankel function `h2v` itself.

    Returns
    -------
    scalar or ndarray
        Values of the derivative of the Hankel function.

    See Also
    --------
    hankel2

    Notes
    -----
    The derivative is computed using the relation DLFM 10.6.7 [2]_.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 5.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    .. [2] NIST Digital Library of Mathematical Functions.
           https://dlmf.nist.gov/10.6.E7

    Examples
    --------
    Compute the Hankel function of the second kind of order 0 and
    its first two derivatives at 1.

    >>> from scipy.special import h2vp
    >>> h2vp(0, 1, 0), h2vp(0, 1, 1), h2vp(0, 1, 2)
    ((0.7651976865579664-0.088256964215677j),
     (-0.44005058574493355-0.7812128213002889j),
     (-0.3251471008130329+0.8694697855159659j))

    Compute the first derivative of the Hankel function of the second kind
    for several orders at 1 by providing an array for `v`.

    >>> h2vp([0, 1, 2], 1, 1)
    array([-0.44005059-0.78121282j,  0.3251471 -0.86946979j,
           0.21024362-2.52015239j])

    Compute the first derivative of the Hankel function of the second kind
    of order 0 at several points by providing an array for `z`.

    >>> import numpy as np
    >>> points = np.array([0.5, 1.5, 3.])
    >>> h2vp(0, points, 1)
    array([-0.24226846-1.47147239j, -0.55793651-0.41230863j,
           -0.33905896+0.32467442j])
    """
    n = _nonneg_int_or_fail(n, 'n')
    if n == 0:
        return hankel2(v, z)
    else:
        return _bessel_diff_formula(v, z, n, hankel2, -1)


def riccati_jn(n, x):
    r"""Compute Ricatti-Bessel function of the first kind and its derivative.

    The Ricatti-Bessel function of the first kind is defined as :math:`x
    j_n(x)`, where :math:`j_n` is the spherical Bessel function of the first
    kind of order :math:`n`.

    This function computes the value and first derivative of the
    Ricatti-Bessel function for all orders up to and including `n`.

    Parameters
    ----------
    n : int
        Maximum order of function to compute
    x : float
        Argument at which to evaluate

    Returns
    -------
    jn : ndarray
        Value of j0(x), ..., jn(x)
    jnp : ndarray
        First derivative j0'(x), ..., jn'(x)

    Notes
    -----
    The computation is carried out via backward recurrence, using the
    relation DLMF 10.51.1 [2]_.

    Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
    Jin [1]_.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
    .. [2] NIST Digital Library of Mathematical Functions.
           https://dlmf.nist.gov/10.51.E1

    """
    if not (isscalar(n) and isscalar(x)):
        raise ValueError("arguments must be scalars.")
    n = _nonneg_int_or_fail(n, 'n', strict=False)
    if (n == 0):
        n1 = 1
    else:
        n1 = n
    nm, jn, jnp = _specfun.rctj(n1, x)
    return jn[:(n+1)], jnp[:(n+1)]


def riccati_yn(n, x):
    """Compute Ricatti-Bessel function of the second kind and its derivative.

    The Ricatti-Bessel function of the second kind is defined as :math:`x
    y_n(x)`, where :math:`y_n` is the spherical Bessel function of the second
    kind of order :math:`n`.

    This function computes the value and first derivative of the function for
    all orders up to and including `n`.

    Parameters
    ----------
    n : int
        Maximum order of function to compute
    x : float
        Argument at which to evaluate

    Returns
    -------
    yn : ndarray
        Value of y0(x), ..., yn(x)
    ynp : ndarray
        First derivative y0'(x), ..., yn'(x)

    Notes
    -----
    The computation is carried out via ascending recurrence, using the
    relation DLMF 10.51.1 [2]_.

    Wrapper for a Fortran routine created by Shanjie Zhang and Jianming
    Jin [1]_.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
    .. [2] NIST Digital Library of Mathematical Functions.
           https://dlmf.nist.gov/10.51.E1

    """
    if not (isscalar(n) and isscalar(x)):
        raise ValueError("arguments must be scalars.")
    n = _nonneg_int_or_fail(n, 'n', strict=False)
    if (n == 0):
        n1 = 1
    else:
        n1 = n
    nm, jn, jnp = _specfun.rcty(n1, x)
    return jn[:(n+1)], jnp[:(n+1)]


def erf_zeros(nt):
    """Compute the first nt zero in the first quadrant, ordered by absolute value.

    Zeros in the other quadrants can be obtained by using the symmetries
    erf(-z) = erf(z) and erf(conj(z)) = conj(erf(z)).


    Parameters
    ----------
    nt : int
        The number of zeros to compute

    Returns
    -------
    The locations of the zeros of erf : ndarray (complex)
        Complex values at which zeros of erf(z)

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    >>> from scipy import special
    >>> special.erf_zeros(1)
    array([1.45061616+1.880943j])

    Check that erf is (close to) zero for the value returned by erf_zeros

    >>> special.erf(special.erf_zeros(1))
    array([4.95159469e-14-1.16407394e-16j])

    """
    if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt):
        raise ValueError("Argument must be positive scalar integer.")
    return _specfun.cerzo(nt)


def fresnelc_zeros(nt):
    """Compute nt complex zeros of cosine Fresnel integral C(z).

    Parameters
    ----------
    nt : int
        Number of zeros to compute

    Returns
    -------
    fresnelc_zeros: ndarray
        Zeros of the cosine Fresnel integral

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt):
        raise ValueError("Argument must be positive scalar integer.")
    return _specfun.fcszo(1, nt)


def fresnels_zeros(nt):
    """Compute nt complex zeros of sine Fresnel integral S(z).

    Parameters
    ----------
    nt : int
        Number of zeros to compute

    Returns
    -------
    fresnels_zeros: ndarray
        Zeros of the sine Fresnel integral

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt):
        raise ValueError("Argument must be positive scalar integer.")
    return _specfun.fcszo(2, nt)


def fresnel_zeros(nt):
    """Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z).

    Parameters
    ----------
    nt : int
        Number of zeros to compute

    Returns
    -------
    zeros_sine: ndarray
        Zeros of the sine Fresnel integral
    zeros_cosine : ndarray
        Zeros of the cosine Fresnel integral

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt):
        raise ValueError("Argument must be positive scalar integer.")
    return _specfun.fcszo(2, nt), _specfun.fcszo(1, nt)


def assoc_laguerre(x, n, k=0.0):
    """Compute the generalized (associated) Laguerre polynomial of degree n and order k.

    The polynomial :math:`L^{(k)}_n(x)` is orthogonal over ``[0, inf)``,
    with weighting function ``exp(-x) * x**k`` with ``k > -1``.

    Parameters
    ----------
    x : float or ndarray
        Points where to evaluate the Laguerre polynomial
    n : int
        Degree of the Laguerre polynomial
    k : int
        Order of the Laguerre polynomial

    Returns
    -------
    assoc_laguerre: float or ndarray
        Associated laguerre polynomial values

    Notes
    -----
    `assoc_laguerre` is a simple wrapper around `eval_genlaguerre`, with
    reversed argument order ``(x, n, k=0.0) --> (n, k, x)``.

    """
    return _ufuncs.eval_genlaguerre(n, k, x)


digamma = psi


def polygamma(n, x):
    r"""Polygamma functions.

    Defined as :math:`\psi^{(n)}(x)` where :math:`\psi` is the
    `digamma` function. See [dlmf]_ for details.

    Parameters
    ----------
    n : array_like
        The order of the derivative of the digamma function; must be
        integral
    x : array_like
        Real valued input

    Returns
    -------
    ndarray
        Function results

    See Also
    --------
    digamma

    References
    ----------
    .. [dlmf] NIST, Digital Library of Mathematical Functions,
        https://dlmf.nist.gov/5.15

    Examples
    --------
    >>> from scipy import special
    >>> x = [2, 3, 25.5]
    >>> special.polygamma(1, x)
    array([ 0.64493407,  0.39493407,  0.03999467])
    >>> special.polygamma(0, x) == special.psi(x)
    array([ True,  True,  True], dtype=bool)

    """
    n, x = asarray(n), asarray(x)
    fac2 = (-1.0)**(n+1) * gamma(n+1.0) * zeta(n+1, x)
    return where(n == 0, psi(x), fac2)


def mathieu_even_coef(m, q):
    r"""Fourier coefficients for even Mathieu and modified Mathieu functions.

    The Fourier series of the even solutions of the Mathieu differential
    equation are of the form

    .. math:: \mathrm{ce}_{2n}(z, q) = \sum_{k=0}^{\infty} A_{(2n)}^{(2k)} \cos 2kz

    .. math:: \mathrm{ce}_{2n+1}(z, q) =
              \sum_{k=0}^{\infty} A_{(2n+1)}^{(2k+1)} \cos (2k+1)z

    This function returns the coefficients :math:`A_{(2n)}^{(2k)}` for even
    input m=2n, and the coefficients :math:`A_{(2n+1)}^{(2k+1)}` for odd input
    m=2n+1.

    Parameters
    ----------
    m : int
        Order of Mathieu functions.  Must be non-negative.
    q : float (>=0)
        Parameter of Mathieu functions.  Must be non-negative.

    Returns
    -------
    Ak : ndarray
        Even or odd Fourier coefficients, corresponding to even or odd m.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
    .. [2] NIST Digital Library of Mathematical Functions
           https://dlmf.nist.gov/28.4#i

    """
    if not (isscalar(m) and isscalar(q)):
        raise ValueError("m and q must be scalars.")
    if (q < 0):
        raise ValueError("q >=0")
    if (m != floor(m)) or (m < 0):
        raise ValueError("m must be an integer >=0.")

    if (q <= 1):
        qm = 7.5 + 56.1*sqrt(q) - 134.7*q + 90.7*sqrt(q)*q
    else:
        qm = 17.0 + 3.1*sqrt(q) - .126*q + .0037*sqrt(q)*q
    km = int(qm + 0.5*m)
    if km > 251:
        warnings.warn("Too many predicted coefficients.", RuntimeWarning, stacklevel=2)
    kd = 1
    m = int(floor(m))
    if m % 2:
        kd = 2

    a = mathieu_a(m, q)
    fc = _specfun.fcoef(kd, m, q, a)
    return fc[:km]


def mathieu_odd_coef(m, q):
    r"""Fourier coefficients for even Mathieu and modified Mathieu functions.

    The Fourier series of the odd solutions of the Mathieu differential
    equation are of the form

    .. math:: \mathrm{se}_{2n+1}(z, q) =
              \sum_{k=0}^{\infty} B_{(2n+1)}^{(2k+1)} \sin (2k+1)z

    .. math:: \mathrm{se}_{2n+2}(z, q) =
              \sum_{k=0}^{\infty} B_{(2n+2)}^{(2k+2)} \sin (2k+2)z

    This function returns the coefficients :math:`B_{(2n+2)}^{(2k+2)}` for even
    input m=2n+2, and the coefficients :math:`B_{(2n+1)}^{(2k+1)}` for odd
    input m=2n+1.

    Parameters
    ----------
    m : int
        Order of Mathieu functions.  Must be non-negative.
    q : float (>=0)
        Parameter of Mathieu functions.  Must be non-negative.

    Returns
    -------
    Bk : ndarray
        Even or odd Fourier coefficients, corresponding to even or odd m.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not (isscalar(m) and isscalar(q)):
        raise ValueError("m and q must be scalars.")
    if (q < 0):
        raise ValueError("q >=0")
    if (m != floor(m)) or (m <= 0):
        raise ValueError("m must be an integer > 0")

    if (q <= 1):
        qm = 7.5 + 56.1*sqrt(q) - 134.7*q + 90.7*sqrt(q)*q
    else:
        qm = 17.0 + 3.1*sqrt(q) - .126*q + .0037*sqrt(q)*q
    km = int(qm + 0.5*m)
    if km > 251:
        warnings.warn("Too many predicted coefficients.", RuntimeWarning, stacklevel=2)
    kd = 4
    m = int(floor(m))
    if m % 2:
        kd = 3

    b = mathieu_b(m, q)
    fc = _specfun.fcoef(kd, m, q, b)
    return fc[:km]


def lpmn(m, n, z):
    """Sequence of associated Legendre functions of the first kind.

    Computes the associated Legendre function of the first kind of order m and
    degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``.
    Returns two arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and
    ``Pmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``.

    This function takes a real argument ``z``. For complex arguments ``z``
    use clpmn instead.

    Parameters
    ----------
    m : int
       ``|m| <= n``; the order of the Legendre function.
    n : int
       where ``n >= 0``; the degree of the Legendre function.  Often
       called ``l`` (lower case L) in descriptions of the associated
       Legendre function
    z : float
        Input value.

    Returns
    -------
    Pmn_z : (m+1, n+1) array
       Values for all orders 0..m and degrees 0..n
    Pmn_d_z : (m+1, n+1) array
       Derivatives for all orders 0..m and degrees 0..n

    See Also
    --------
    clpmn: associated Legendre functions of the first kind for complex z

    Notes
    -----
    In the interval (-1, 1), Ferrer's function of the first kind is
    returned. The phase convention used for the intervals (1, inf)
    and (-inf, -1) is such that the result is always real.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
    .. [2] NIST Digital Library of Mathematical Functions
           https://dlmf.nist.gov/14.3

    """
    if not isscalar(m) or (abs(m) > n):
        raise ValueError("m must be <= n.")
    if not isscalar(n) or (n < 0):
        raise ValueError("n must be a non-negative integer.")
    if not isscalar(z):
        raise ValueError("z must be scalar.")
    if iscomplex(z):
        raise ValueError("Argument must be real. Use clpmn instead.")
    if (m < 0):
        mp = -m
        mf, nf = mgrid[0:mp+1, 0:n+1]
        with _ufuncs.errstate(all='ignore'):
            if abs(z) < 1:
                # Ferrer function; DLMF 14.9.3
                fixarr = where(mf > nf, 0.0,
                               (-1)**mf * gamma(nf-mf+1) / gamma(nf+mf+1))
            else:
                # Match to clpmn; DLMF 14.9.13
                fixarr = where(mf > nf, 0.0, gamma(nf-mf+1) / gamma(nf+mf+1))
    else:
        mp = m
    p, pd = _specfun.lpmn(mp, n, z)
    if (m < 0):
        p = p * fixarr
        pd = pd * fixarr
    return p, pd


def clpmn(m, n, z, type=3):
    """Associated Legendre function of the first kind for complex arguments.

    Computes the associated Legendre function of the first kind of order m and
    degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``.
    Returns two arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and
    ``Pmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``.

    Parameters
    ----------
    m : int
       ``|m| <= n``; the order of the Legendre function.
    n : int
       where ``n >= 0``; the degree of the Legendre function.  Often
       called ``l`` (lower case L) in descriptions of the associated
       Legendre function
    z : float or complex
        Input value.
    type : int, optional
       takes values 2 or 3
       2: cut on the real axis ``|x| > 1``
       3: cut on the real axis ``-1 < x < 1`` (default)

    Returns
    -------
    Pmn_z : (m+1, n+1) array
       Values for all orders ``0..m`` and degrees ``0..n``
    Pmn_d_z : (m+1, n+1) array
       Derivatives for all orders ``0..m`` and degrees ``0..n``

    See Also
    --------
    lpmn: associated Legendre functions of the first kind for real z

    Notes
    -----
    By default, i.e. for ``type=3``, phase conventions are chosen according
    to [1]_ such that the function is analytic. The cut lies on the interval
    (-1, 1). Approaching the cut from above or below in general yields a phase
    factor with respect to Ferrer's function of the first kind
    (cf. `lpmn`).

    For ``type=2`` a cut at ``|x| > 1`` is chosen. Approaching the real values
    on the interval (-1, 1) in the complex plane yields Ferrer's function
    of the first kind.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
    .. [2] NIST Digital Library of Mathematical Functions
           https://dlmf.nist.gov/14.21

    """
    if not isscalar(m) or (abs(m) > n):
        raise ValueError("m must be <= n.")
    if not isscalar(n) or (n < 0):
        raise ValueError("n must be a non-negative integer.")
    if not isscalar(z):
        raise ValueError("z must be scalar.")
    if not (type == 2 or type == 3):
        raise ValueError("type must be either 2 or 3.")
    if (m < 0):
        mp = -m
        mf, nf = mgrid[0:mp+1, 0:n+1]
        with _ufuncs.errstate(all='ignore'):
            if type == 2:
                fixarr = where(mf > nf, 0.0,
                               (-1)**mf * gamma(nf-mf+1) / gamma(nf+mf+1))
            else:
                fixarr = where(mf > nf, 0.0, gamma(nf-mf+1) / gamma(nf+mf+1))
    else:
        mp = m
    p, pd = _specfun.clpmn(mp, n, z, type)
    if (m < 0):
        p = p * fixarr
        pd = pd * fixarr
    return p, pd


def lqmn(m, n, z):
    """Sequence of associated Legendre functions of the second kind.

    Computes the associated Legendre function of the second kind of order m and
    degree n, ``Qmn(z)`` = :math:`Q_n^m(z)`, and its derivative, ``Qmn'(z)``.
    Returns two arrays of size ``(m+1, n+1)`` containing ``Qmn(z)`` and
    ``Qmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``.

    Parameters
    ----------
    m : int
       ``|m| <= n``; the order of the Legendre function.
    n : int
       where ``n >= 0``; the degree of the Legendre function.  Often
       called ``l`` (lower case L) in descriptions of the associated
       Legendre function
    z : complex
        Input value.

    Returns
    -------
    Qmn_z : (m+1, n+1) array
       Values for all orders 0..m and degrees 0..n
    Qmn_d_z : (m+1, n+1) array
       Derivatives for all orders 0..m and degrees 0..n

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not isscalar(m) or (m < 0):
        raise ValueError("m must be a non-negative integer.")
    if not isscalar(n) or (n < 0):
        raise ValueError("n must be a non-negative integer.")
    if not isscalar(z):
        raise ValueError("z must be scalar.")
    m = int(m)
    n = int(n)

    # Ensure neither m nor n == 0
    mm = max(1, m)
    nn = max(1, n)

    if iscomplex(z):
        q, qd = _specfun.clqmn(mm, nn, z)
    else:
        q, qd = _specfun.lqmn(mm, nn, z)
    return q[:(m+1), :(n+1)], qd[:(m+1), :(n+1)]


def bernoulli(n):
    """Bernoulli numbers B0..Bn (inclusive).

    Parameters
    ----------
    n : int
        Indicated the number of terms in the Bernoulli series to generate.

    Returns
    -------
    ndarray
        The Bernoulli numbers ``[B(0), B(1), ..., B(n)]``.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
    .. [2] "Bernoulli number", Wikipedia, https://en.wikipedia.org/wiki/Bernoulli_number

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.special import bernoulli, zeta
    >>> bernoulli(4)
    array([ 1.        , -0.5       ,  0.16666667,  0.        , -0.03333333])

    The Wikipedia article ([2]_) points out the relationship between the
    Bernoulli numbers and the zeta function, ``B_n^+ = -n * zeta(1 - n)``
    for ``n > 0``:

    >>> n = np.arange(1, 5)
    >>> -n * zeta(1 - n)
    array([ 0.5       ,  0.16666667, -0.        , -0.03333333])

    Note that, in the notation used in the wikipedia article,
    `bernoulli` computes ``B_n^-`` (i.e. it used the convention that
    ``B_1`` is -1/2).  The relation given above is for ``B_n^+``, so the
    sign of 0.5 does not match the output of ``bernoulli(4)``.

    """
    if not isscalar(n) or (n < 0):
        raise ValueError("n must be a non-negative integer.")
    n = int(n)
    if (n < 2):
        n1 = 2
    else:
        n1 = n
    return _specfun.bernob(int(n1))[:(n+1)]


def euler(n):
    """Euler numbers E(0), E(1), ..., E(n).

    The Euler numbers [1]_ are also known as the secant numbers.

    Because ``euler(n)`` returns floating point values, it does not give
    exact values for large `n`.  The first inexact value is E(22).

    Parameters
    ----------
    n : int
        The highest index of the Euler number to be returned.

    Returns
    -------
    ndarray
        The Euler numbers [E(0), E(1), ..., E(n)].
        The odd Euler numbers, which are all zero, are included.

    References
    ----------
    .. [1] Sequence A122045, The On-Line Encyclopedia of Integer Sequences,
           https://oeis.org/A122045
    .. [2] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.special import euler
    >>> euler(6)
    array([  1.,   0.,  -1.,   0.,   5.,   0., -61.])

    >>> euler(13).astype(np.int64)
    array([      1,       0,      -1,       0,       5,       0,     -61,
                 0,    1385,       0,  -50521,       0, 2702765,       0])

    >>> euler(22)[-1]  # Exact value of E(22) is -69348874393137901.
    -69348874393137976.0

    """
    if not isscalar(n) or (n < 0):
        raise ValueError("n must be a non-negative integer.")
    n = int(n)
    if (n < 2):
        n1 = 2
    else:
        n1 = n
    return _specfun.eulerb(n1)[:(n+1)]


def lpn(n, z):
    """Legendre function of the first kind.

    Compute sequence of Legendre functions of the first kind (polynomials),
    Pn(z) and derivatives for all degrees from 0 to n (inclusive).

    See also special.legendre for polynomial class.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not (isscalar(n) and isscalar(z)):
        raise ValueError("arguments must be scalars.")
    n = _nonneg_int_or_fail(n, 'n', strict=False)
    if (n < 1):
        n1 = 1
    else:
        n1 = n
    if iscomplex(z):
        pn, pd = _specfun.clpn(n1, z)
    else:
        pn, pd = _specfun.lpn(n1, z)
    return pn[:(n+1)], pd[:(n+1)]


def lqn(n, z):
    """Legendre function of the second kind.

    Compute sequence of Legendre functions of the second kind, Qn(z) and
    derivatives for all degrees from 0 to n (inclusive).

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not (isscalar(n) and isscalar(z)):
        raise ValueError("arguments must be scalars.")
    n = _nonneg_int_or_fail(n, 'n', strict=False)
    if (n < 1):
        n1 = 1
    else:
        n1 = n
    if iscomplex(z):
        qn, qd = _specfun.clqn(n1, z)
    else:
        qn, qd = _specfun.lqnb(n1, z)
    return qn[:(n+1)], qd[:(n+1)]


def ai_zeros(nt):
    """
    Compute `nt` zeros and values of the Airy function Ai and its derivative.

    Computes the first `nt` zeros, `a`, of the Airy function Ai(x);
    first `nt` zeros, `ap`, of the derivative of the Airy function Ai'(x);
    the corresponding values Ai(a');
    and the corresponding values Ai'(a).

    Parameters
    ----------
    nt : int
        Number of zeros to compute

    Returns
    -------
    a : ndarray
        First `nt` zeros of Ai(x)
    ap : ndarray
        First `nt` zeros of Ai'(x)
    ai : ndarray
        Values of Ai(x) evaluated at first `nt` zeros of Ai'(x)
    aip : ndarray
        Values of Ai'(x) evaluated at first `nt` zeros of Ai(x)

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    >>> from scipy import special
    >>> a, ap, ai, aip = special.ai_zeros(3)
    >>> a
    array([-2.33810741, -4.08794944, -5.52055983])
    >>> ap
    array([-1.01879297, -3.24819758, -4.82009921])
    >>> ai
    array([ 0.53565666, -0.41901548,  0.38040647])
    >>> aip
    array([ 0.70121082, -0.80311137,  0.86520403])

    """
    kf = 1
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("nt must be a positive integer scalar.")
    return _specfun.airyzo(nt, kf)


def bi_zeros(nt):
    """
    Compute `nt` zeros and values of the Airy function Bi and its derivative.

    Computes the first `nt` zeros, b, of the Airy function Bi(x);
    first `nt` zeros, b', of the derivative of the Airy function Bi'(x);
    the corresponding values Bi(b');
    and the corresponding values Bi'(b).

    Parameters
    ----------
    nt : int
        Number of zeros to compute

    Returns
    -------
    b : ndarray
        First `nt` zeros of Bi(x)
    bp : ndarray
        First `nt` zeros of Bi'(x)
    bi : ndarray
        Values of Bi(x) evaluated at first `nt` zeros of Bi'(x)
    bip : ndarray
        Values of Bi'(x) evaluated at first `nt` zeros of Bi(x)

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    Examples
    --------
    >>> from scipy import special
    >>> b, bp, bi, bip = special.bi_zeros(3)
    >>> b
    array([-1.17371322, -3.2710933 , -4.83073784])
    >>> bp
    array([-2.29443968, -4.07315509, -5.51239573])
    >>> bi
    array([-0.45494438,  0.39652284, -0.36796916])
    >>> bip
    array([ 0.60195789, -0.76031014,  0.83699101])

    """
    kf = 2
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("nt must be a positive integer scalar.")
    return _specfun.airyzo(nt, kf)


def lmbda(v, x):
    r"""Jahnke-Emden Lambda function, Lambdav(x).

    This function is defined as [2]_,

    .. math:: \Lambda_v(x) = \Gamma(v+1) \frac{J_v(x)}{(x/2)^v},

    where :math:`\Gamma` is the gamma function and :math:`J_v` is the
    Bessel function of the first kind.

    Parameters
    ----------
    v : float
        Order of the Lambda function
    x : float
        Value at which to evaluate the function and derivatives

    Returns
    -------
    vl : ndarray
        Values of Lambda_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
    dl : ndarray
        Derivatives Lambda_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html
    .. [2] Jahnke, E. and Emde, F. "Tables of Functions with Formulae and
           Curves" (4th ed.), Dover, 1945
    """
    if not (isscalar(v) and isscalar(x)):
        raise ValueError("arguments must be scalars.")
    if (v < 0):
        raise ValueError("argument must be > 0.")
    n = int(v)
    v0 = v - n
    if (n < 1):
        n1 = 1
    else:
        n1 = n
    v1 = n1 + v0
    if (v != floor(v)):
        vm, vl, dl = _specfun.lamv(v1, x)
    else:
        vm, vl, dl = _specfun.lamn(v1, x)
    return vl[:(n+1)], dl[:(n+1)]


def pbdv_seq(v, x):
    """Parabolic cylinder functions Dv(x) and derivatives.

    Parameters
    ----------
    v : float
        Order of the parabolic cylinder function
    x : float
        Value at which to evaluate the function and derivatives

    Returns
    -------
    dv : ndarray
        Values of D_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
    dp : ndarray
        Derivatives D_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 13.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not (isscalar(v) and isscalar(x)):
        raise ValueError("arguments must be scalars.")
    n = int(v)
    v0 = v-n
    if (n < 1):
        n1 = 1
    else:
        n1 = n
    v1 = n1 + v0
    dv, dp, pdf, pdd = _specfun.pbdv(v1, x)
    return dv[:n1+1], dp[:n1+1]


def pbvv_seq(v, x):
    """Parabolic cylinder functions Vv(x) and derivatives.

    Parameters
    ----------
    v : float
        Order of the parabolic cylinder function
    x : float
        Value at which to evaluate the function and derivatives

    Returns
    -------
    dv : ndarray
        Values of V_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.
    dp : ndarray
        Derivatives V_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 13.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not (isscalar(v) and isscalar(x)):
        raise ValueError("arguments must be scalars.")
    n = int(v)
    v0 = v-n
    if (n <= 1):
        n1 = 1
    else:
        n1 = n
    v1 = n1 + v0
    dv, dp, pdf, pdd = _specfun.pbvv(v1, x)
    return dv[:n1+1], dp[:n1+1]


def pbdn_seq(n, z):
    """Parabolic cylinder functions Dn(z) and derivatives.

    Parameters
    ----------
    n : int
        Order of the parabolic cylinder function
    z : complex
        Value at which to evaluate the function and derivatives

    Returns
    -------
    dv : ndarray
        Values of D_i(z), for i=0, ..., i=n.
    dp : ndarray
        Derivatives D_i'(z), for i=0, ..., i=n.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996, chapter 13.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not (isscalar(n) and isscalar(z)):
        raise ValueError("arguments must be scalars.")
    if (floor(n) != n):
        raise ValueError("n must be an integer.")
    if (abs(n) <= 1):
        n1 = 1
    else:
        n1 = n
    cpb, cpd = _specfun.cpbdn(n1, z)
    return cpb[:n1+1], cpd[:n1+1]


def ber_zeros(nt):
    """Compute nt zeros of the Kelvin function ber.

    Parameters
    ----------
    nt : int
        Number of zeros to compute. Must be positive.

    Returns
    -------
    ndarray
        First `nt` zeros of the Kelvin function.

    See Also
    --------
    ber

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("nt must be positive integer scalar.")
    return _specfun.klvnzo(nt, 1)


def bei_zeros(nt):
    """Compute nt zeros of the Kelvin function bei.

    Parameters
    ----------
    nt : int
        Number of zeros to compute. Must be positive.

    Returns
    -------
    ndarray
        First `nt` zeros of the Kelvin function.

    See Also
    --------
    bei

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("nt must be positive integer scalar.")
    return _specfun.klvnzo(nt, 2)


def ker_zeros(nt):
    """Compute nt zeros of the Kelvin function ker.

    Parameters
    ----------
    nt : int
        Number of zeros to compute. Must be positive.

    Returns
    -------
    ndarray
        First `nt` zeros of the Kelvin function.

    See Also
    --------
    ker

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("nt must be positive integer scalar.")
    return _specfun.klvnzo(nt, 3)


def kei_zeros(nt):
    """Compute nt zeros of the Kelvin function kei.

    Parameters
    ----------
    nt : int
        Number of zeros to compute. Must be positive.

    Returns
    -------
    ndarray
        First `nt` zeros of the Kelvin function.

    See Also
    --------
    kei

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("nt must be positive integer scalar.")
    return _specfun.klvnzo(nt, 4)


def berp_zeros(nt):
    """Compute nt zeros of the derivative of the Kelvin function ber.

    Parameters
    ----------
    nt : int
        Number of zeros to compute. Must be positive.

    Returns
    -------
    ndarray
        First `nt` zeros of the derivative of the Kelvin function.

    See Also
    --------
    ber, berp

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("nt must be positive integer scalar.")
    return _specfun.klvnzo(nt, 5)


def beip_zeros(nt):
    """Compute nt zeros of the derivative of the Kelvin function bei.

    Parameters
    ----------
    nt : int
        Number of zeros to compute. Must be positive.

    Returns
    -------
    ndarray
        First `nt` zeros of the derivative of the Kelvin function.

    See Also
    --------
    bei, beip

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("nt must be positive integer scalar.")
    return _specfun.klvnzo(nt, 6)


def kerp_zeros(nt):
    """Compute nt zeros of the derivative of the Kelvin function ker.

    Parameters
    ----------
    nt : int
        Number of zeros to compute. Must be positive.

    Returns
    -------
    ndarray
        First `nt` zeros of the derivative of the Kelvin function.

    See Also
    --------
    ker, kerp

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("nt must be positive integer scalar.")
    return _specfun.klvnzo(nt, 7)


def keip_zeros(nt):
    """Compute nt zeros of the derivative of the Kelvin function kei.

    Parameters
    ----------
    nt : int
        Number of zeros to compute. Must be positive.

    Returns
    -------
    ndarray
        First `nt` zeros of the derivative of the Kelvin function.

    See Also
    --------
    kei, keip

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("nt must be positive integer scalar.")
    return _specfun.klvnzo(nt, 8)


def kelvin_zeros(nt):
    """Compute nt zeros of all Kelvin functions.

    Returned in a length-8 tuple of arrays of length nt.  The tuple contains
    the arrays of zeros of (ber, bei, ker, kei, ber', bei', ker', kei').

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0):
        raise ValueError("nt must be positive integer scalar.")
    return (_specfun.klvnzo(nt, 1),
            _specfun.klvnzo(nt, 2),
            _specfun.klvnzo(nt, 3),
            _specfun.klvnzo(nt, 4),
            _specfun.klvnzo(nt, 5),
            _specfun.klvnzo(nt, 6),
            _specfun.klvnzo(nt, 7),
            _specfun.klvnzo(nt, 8))


def pro_cv_seq(m, n, c):
    """Characteristic values for prolate spheroidal wave functions.

    Compute a sequence of characteristic values for the prolate
    spheroidal wave functions for mode m and n'=m..n and spheroidal
    parameter c.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not (isscalar(m) and isscalar(n) and isscalar(c)):
        raise ValueError("Arguments must be scalars.")
    if (n != floor(n)) or (m != floor(m)):
        raise ValueError("Modes must be integers.")
    if (n-m > 199):
        raise ValueError("Difference between n and m is too large.")
    maxL = n-m+1
    return _specfun.segv(m, n, c, 1)[1][:maxL]


def obl_cv_seq(m, n, c):
    """Characteristic values for oblate spheroidal wave functions.

    Compute a sequence of characteristic values for the oblate
    spheroidal wave functions for mode m and n'=m..n and spheroidal
    parameter c.

    References
    ----------
    .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
           Functions", John Wiley and Sons, 1996.
           https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

    """
    if not (isscalar(m) and isscalar(n) and isscalar(c)):
        raise ValueError("Arguments must be scalars.")
    if (n != floor(n)) or (m != floor(m)):
        raise ValueError("Modes must be integers.")
    if (n-m > 199):
        raise ValueError("Difference between n and m is too large.")
    maxL = n-m+1
    return _specfun.segv(m, n, c, -1)[1][:maxL]


@_deprecate_positional_args(version="1.14")
def comb(N, k, *, exact=False, repetition=False, legacy=_NoValue):
    """The number of combinations of N things taken k at a time.

    This is often expressed as "N choose k".

    Parameters
    ----------
    N : int, ndarray
        Number of things.
    k : int, ndarray
        Number of elements taken.
    exact : bool, optional
        For integers, if `exact` is False, then floating point precision is
        used, otherwise the result is computed exactly. For non-integers, if
        `exact` is True, is disregarded.
    repetition : bool, optional
        If `repetition` is True, then the number of combinations with
        repetition is computed.
    legacy : bool, optional
        If `legacy` is True and `exact` is True, then non-integral arguments
        are cast to ints; if `legacy` is False, the result for non-integral
        arguments is unaffected by the value of `exact`.

        .. deprecated:: 1.9.0
            Using `legacy` is deprecated and will removed by
            Scipy 1.14.0. If you want to keep the legacy behaviour, cast
            your inputs directly, e.g.
            ``comb(int(your_N), int(your_k), exact=True)``.

    Returns
    -------
    val : int, float, ndarray
        The total number of combinations.

    See Also
    --------
    binom : Binomial coefficient considered as a function of two real
            variables.

    Notes
    -----
    - Array arguments accepted only for exact=False case.
    - If N < 0, or k < 0, then 0 is returned.
    - If k > N and repetition=False, then 0 is returned.

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.special import comb
    >>> k = np.array([3, 4])
    >>> n = np.array([10, 10])
    >>> comb(n, k, exact=False)
    array([ 120.,  210.])
    >>> comb(10, 3, exact=True)
    120
    >>> comb(10, 3, exact=True, repetition=True)
    220

    """
    if legacy is not _NoValue:
        warnings.warn(
            "Using 'legacy' keyword is deprecated and will be removed by "
            "Scipy 1.14.0. If you want to keep the legacy behaviour, cast "
            "your inputs directly, e.g. "
            "'comb(int(your_N), int(your_k), exact=True)'.",
            DeprecationWarning,
            stacklevel=2
        )
    if repetition:
        return comb(N + k - 1, k, exact=exact, legacy=legacy)
    if exact:
        if int(N) == N and int(k) == k:
            # _comb_int casts inputs to integers, which is safe & intended here
            return _comb_int(N, k)
        elif legacy:
            # here at least one number is not an integer; legacy behavior uses
            # lossy casts to int
            return _comb_int(N, k)
        # otherwise, we disregard `exact=True`; it makes no sense for
        # non-integral arguments
        return comb(N, k)
    else:
        k, N = asarray(k), asarray(N)
        cond = (k <= N) & (N >= 0) & (k >= 0)
        vals = binom(N, k)
        if isinstance(vals, np.ndarray):
            vals[~cond] = 0
        elif not cond:
            vals = np.float64(0)
        return vals


def perm(N, k, exact=False):
    """Permutations of N things taken k at a time, i.e., k-permutations of N.

    It's also known as "partial permutations".

    Parameters
    ----------
    N : int, ndarray
        Number of things.
    k : int, ndarray
        Number of elements taken.
    exact : bool, optional
        If `exact` is False, then floating point precision is used, otherwise
        exact long integer is computed.

    Returns
    -------
    val : int, ndarray
        The number of k-permutations of N.

    Notes
    -----
    - Array arguments accepted only for exact=False case.
    - If k > N, N < 0, or k < 0, then a 0 is returned.

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.special import perm
    >>> k = np.array([3, 4])
    >>> n = np.array([10, 10])
    >>> perm(n, k)
    array([  720.,  5040.])
    >>> perm(10, 3, exact=True)
    720

    """
    if exact:
        if (k > N) or (N < 0) or (k < 0):
            return 0
        val = 1
        for i in range(N - k + 1, N + 1):
            val *= i
        return val
    else:
        k, N = asarray(k), asarray(N)
        cond = (k <= N) & (N >= 0) & (k >= 0)
        vals = poch(N - k + 1, k)
        if isinstance(vals, np.ndarray):
            vals[~cond] = 0
        elif not cond:
            vals = np.float64(0)
        return vals


# https://stackoverflow.com/a/16327037
def _range_prod(lo, hi, k=1):
    """
    Product of a range of numbers spaced k apart (from hi).

    For k=1, this returns the product of
    lo * (lo+1) * (lo+2) * ... * (hi-2) * (hi-1) * hi
    = hi! / (lo-1)!

    For k>1, it correspond to taking only every k'th number when
    counting down from hi - e.g. 18!!!! = _range_prod(1, 18, 4).

    Breaks into smaller products first for speed:
    _range_prod(2, 9) = ((2*3)*(4*5))*((6*7)*(8*9))
    """
    if lo + k < hi:
        mid = (hi + lo) // 2
        if k > 1:
            # make sure mid is a multiple of k away from hi
            mid = mid - ((mid - hi) % k)
        return _range_prod(lo, mid, k) * _range_prod(mid + k, hi, k)
    elif lo + k == hi:
        return lo * hi
    else:
        return hi


def _factorialx_array_exact(n, k=1):
    """
    Exact computation of factorial for an array.

    The factorials are computed in incremental fashion, by taking
    the sorted unique values of n and multiplying the intervening
    numbers between the different unique values.

    In other words, the factorial for the largest input is only
    computed once, with each other result computed in the process.

    k > 1 corresponds to the multifactorial.
    """
    un = np.unique(n)
    # numpy changed nan-sorting behaviour with 1.21, see numpy/numpy#18070;
    # to unify the behaviour, we remove the nan's here; the respective
    # values will be set separately at the end
    un = un[~np.isnan(un)]

    # Convert to object array if np.int64 can't handle size
    if np.isnan(n).any():
        dt = float
    elif k in _FACTORIALK_LIMITS_64BITS.keys():
        if un[-1] > _FACTORIALK_LIMITS_64BITS[k]:
            # e.g. k=1: 21! > np.iinfo(np.int64).max
            dt = object
        elif un[-1] > _FACTORIALK_LIMITS_32BITS[k]:
            # e.g. k=3: 26!!! > np.iinfo(np.int32).max
            dt = np.int64
        else:
            dt = np.dtype("long")
    else:
        # for k >= 10, we always use object
        dt = object

    out = np.empty_like(n, dtype=dt)

    # Handle invalid/trivial values
    un = un[un > 1]
    out[n < 2] = 1
    out[n < 0] = 0

    # Calculate products of each range of numbers
    # we can only multiply incrementally if the values are k apart;
    # therefore we partition `un` into "lanes", i.e. its residues modulo k
    for lane in range(0, k):
        ul = un[(un % k) == lane] if k > 1 else un
        if ul.size:
            # after np.unique, un resp. ul are sorted, ul[0] is the smallest;
            # cast to python ints to avoid overflow with np.int-types
            val = _range_prod(1, int(ul[0]), k=k)
            out[n == ul[0]] = val
            for i in range(len(ul) - 1):
                # by the filtering above, we have ensured that prev & current
                # are a multiple of k apart
                prev = ul[i]
                current = ul[i + 1]
                # we already multiplied all factors until prev; continue
                # building the full factorial from the following (`prev + 1`);
                # use int() for the same reason as above
                val *= _range_prod(int(prev + 1), int(current), k=k)
                out[n == current] = val

    if np.isnan(n).any():
        out = out.astype(np.float64)
        out[np.isnan(n)] = np.nan
    return out


def _factorialx_array_approx(n, k):
    """
    Calculate approximation to multifactorial for array n and integer k.

    Ensure we only call _factorialx_approx_core where necessary/required.
    """
    result = zeros(n.shape)
    # keep nans as nans
    place(result, np.isnan(n), np.nan)
    # only compute where n >= 0 (excludes nans), everything else is 0
    cond = (n >= 0)
    n_to_compute = extract(cond, n)
    place(result, cond, _factorialx_approx_core(n_to_compute, k=k))
    return result


def _factorialx_approx_core(n, k):
    """
    Core approximation to multifactorial for array n and integer k.
    """
    if k == 1:
        # shortcut for k=1
        result = gamma(n + 1)
        if isinstance(n, np.ndarray):
            # gamma does not maintain 0-dim arrays
            result = np.array(result)
        return result

    n_mod_k = n % k
    # scalar case separately, unified handling would be inefficient for arrays;
    # don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below
    if not isinstance(n, np.ndarray):
        return (
            np.power(k, (n - n_mod_k) / k)
            * gamma(n / k + 1) / gamma(n_mod_k / k + 1)
            * max(n_mod_k, 1)
        )

    # factor that's independent of the residue class (see factorialk docstring)
    result = np.power(k, n / k) * gamma(n / k + 1)
    # factor dependent on residue r (for `r=0` it's 1, so we skip `r=0`
    # below and thus also avoid evaluating `max(r, 1)`)
    def corr(k, r): return np.power(k, -r / k) / gamma(r / k + 1) * r
    for r in np.unique(n_mod_k):
        if r == 0:
            continue
        result[n_mod_k == r] *= corr(k, r)
    return result


def factorial(n, exact=False):
    """
    The factorial of a number or array of numbers.

    The factorial of non-negative integer `n` is the product of all
    positive integers less than or equal to `n`::

        n! = n * (n - 1) * (n - 2) * ... * 1

    Parameters
    ----------
    n : int or array_like of ints
        Input values.  If ``n < 0``, the return value is 0.
    exact : bool, optional
        If True, calculate the answer exactly using long integer arithmetic.
        If False, result is approximated in floating point rapidly using the
        `gamma` function.
        Default is False.

    Returns
    -------
    nf : float or int or ndarray
        Factorial of `n`, as integer or float depending on `exact`.

    Notes
    -----
    For arrays with ``exact=True``, the factorial is computed only once, for
    the largest input, with each other result computed in the process.
    The output dtype is increased to ``int64`` or ``object`` if necessary.

    With ``exact=False`` the factorial is approximated using the gamma
    function:

    .. math:: n! = \\Gamma(n+1)

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.special import factorial
    >>> arr = np.array([3, 4, 5])
    >>> factorial(arr, exact=False)
    array([   6.,   24.,  120.])
    >>> factorial(arr, exact=True)
    array([  6,  24, 120])
    >>> factorial(5, exact=True)
    120

    """
    # don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below
    if np.ndim(n) == 0 and not isinstance(n, np.ndarray):
        # scalar cases
        if n is None or np.isnan(n):
            return np.nan
        elif not (np.issubdtype(type(n), np.integer)
                  or np.issubdtype(type(n), np.floating)):
            raise ValueError(
                f"Unsupported datatype for factorial: {type(n)}\n"
                "Permitted data types are integers and floating point numbers"
            )
        elif n < 0:
            return 0
        elif exact and np.issubdtype(type(n), np.integer):
            return math.factorial(n)
        elif exact:
            msg = ("Non-integer values of `n` together with `exact=True` are "
                   "deprecated. Either ensure integer `n` or use `exact=False`.")
            warnings.warn(msg, DeprecationWarning, stacklevel=2)
        return _factorialx_approx_core(n, k=1)

    # arrays & array-likes
    n = asarray(n)
    if n.size == 0:
        # return empty arrays unchanged
        return n
    if not (np.issubdtype(n.dtype, np.integer)
            or np.issubdtype(n.dtype, np.floating)):
        raise ValueError(
            f"Unsupported datatype for factorial: {n.dtype}\n"
            "Permitted data types are integers and floating point numbers"
        )
    if exact and not np.issubdtype(n.dtype, np.integer):
        msg = ("factorial with `exact=True` does not "
               "support non-integral arrays")
        raise ValueError(msg)

    if exact:
        return _factorialx_array_exact(n, k=1)
    return _factorialx_array_approx(n, k=1)


def factorial2(n, exact=False):
    """Double factorial.

    This is the factorial with every second value skipped.  E.g., ``7!! = 7 * 5
    * 3 * 1``.  It can be approximated numerically as::

      n!! = 2 ** (n / 2) * gamma(n / 2 + 1) * sqrt(2 / pi)  n odd
          = 2 ** (n / 2) * gamma(n / 2 + 1)                 n even
          = 2 ** (n / 2) * (n / 2)!                         n even

    Parameters
    ----------
    n : int or array_like
        Calculate ``n!!``.  If ``n < 0``, the return value is 0.
    exact : bool, optional
        The result can be approximated rapidly using the gamma-formula
        above (default).  If `exact` is set to True, calculate the
        answer exactly using integer arithmetic.

    Returns
    -------
    nff : float or int
        Double factorial of `n`, as an int or a float depending on
        `exact`.

    Examples
    --------
    >>> from scipy.special import factorial2
    >>> factorial2(7, exact=False)
    array(105.00000000000001)
    >>> factorial2(7, exact=True)
    105

    """

    # don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below
    if np.ndim(n) == 0 and not isinstance(n, np.ndarray):
        # scalar cases
        if n is None or np.isnan(n):
            return np.nan
        elif not np.issubdtype(type(n), np.integer):
            msg = "factorial2 does not support non-integral scalar arguments"
            raise ValueError(msg)
        elif n < 0:
            return 0
        elif n in {0, 1}:
            return 1
        # general integer case
        if exact:
            return _range_prod(1, n, k=2)
        return _factorialx_approx_core(n, k=2)
    # arrays & array-likes
    n = asarray(n)
    if n.size == 0:
        # return empty arrays unchanged
        return n
    if not np.issubdtype(n.dtype, np.integer):
        raise ValueError("factorial2 does not support non-integral arrays")
    if exact:
        return _factorialx_array_exact(n, k=2)
    return _factorialx_array_approx(n, k=2)


def factorialk(n, k, exact=None):
    """Multifactorial of n of order k, n(!!...!).

    This is the multifactorial of n skipping k values.  For example,

      factorialk(17, 4) = 17!!!! = 17 * 13 * 9 * 5 * 1

    In particular, for any integer ``n``, we have

      factorialk(n, 1) = factorial(n)

      factorialk(n, 2) = factorial2(n)

    Parameters
    ----------
    n : int or array_like
        Calculate multifactorial. If ``n < 0``, the return value is 0.
    k : int
        Order of multifactorial.
    exact : bool, optional
        If exact is set to True, calculate the answer exactly using
        integer arithmetic, otherwise use an approximation (faster,
        but yields floats instead of integers)

        .. warning::
           The default value for ``exact`` will be changed to
           ``False`` in SciPy 1.15.0.

    Returns
    -------
    val : int
        Multifactorial of `n`.

    Examples
    --------
    >>> from scipy.special import factorialk
    >>> factorialk(5, k=1, exact=True)
    120
    >>> factorialk(5, k=3, exact=True)
    10
    >>> factorialk([5, 7, 9], k=3, exact=True)
    array([ 10,  28, 162])
    >>> factorialk([5, 7, 9], k=3, exact=False)
    array([ 10.,  28., 162.])

    Notes
    -----
    While less straight-forward than for the double-factorial, it's possible to
    calculate a general approximation formula of n!(k) by studying ``n`` for a given
    remainder ``r < k`` (thus ``n = m * k + r``, resp. ``r = n % k``), which can be
    put together into something valid for all integer values ``n >= 0`` & ``k > 0``::

      n!(k) = k ** ((n - r)/k) * gamma(n/k + 1) / gamma(r/k + 1) * max(r, 1)

    This is the basis of the approximation when ``exact=False``. Compare also [1].

    References
    ----------
    .. [1] Complex extension to multifactorial
            https://en.wikipedia.org/wiki/Double_factorial#Alternative_extension_of_the_multifactorial
    """
    if not np.issubdtype(type(k), np.integer) or k < 1:
        raise ValueError(f"k must be a positive integer, received: {k}")
    if exact is None:
        msg = (
            "factorialk will default to `exact=False` starting from SciPy "
            "1.15.0. To avoid behaviour changes due to this, explicitly "
            "specify either `exact=False` (faster, returns floats), or the "
            "past default `exact=True` (slower, lossless result as integer)."
        )
        warnings.warn(msg, DeprecationWarning, stacklevel=2)
        exact = True

    helpmsg = ""
    if k in {1, 2}:
        func = "factorial" if k == 1 else "factorial2"
        helpmsg = f"\nYou can try to use {func} instead"

    # don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below
    if np.ndim(n) == 0 and not isinstance(n, np.ndarray):
        # scalar cases
        if n is None or np.isnan(n):
            return np.nan
        elif not np.issubdtype(type(n), np.integer):
            msg = "factorialk does not support non-integral scalar arguments!"
            raise ValueError(msg + helpmsg)
        elif n < 0:
            return 0
        elif n in {0, 1}:
            return 1
        # general integer case
        if exact:
            return _range_prod(1, n, k=k)
        return _factorialx_approx_core(n, k=k)
    # arrays & array-likes
    n = asarray(n)
    if n.size == 0:
        # return empty arrays unchanged
        return n
    if not np.issubdtype(n.dtype, np.integer):
        msg = "factorialk does not support non-integral arrays!"
        raise ValueError(msg + helpmsg)
    if exact:
        return _factorialx_array_exact(n, k=k)
    return _factorialx_array_approx(n, k=k)


def stirling2(N, K, *, exact=False):
    r"""Generate Stirling number(s) of the second kind.

    Stirling numbers of the second kind count the number of ways to
    partition a set with N elements into K non-empty subsets.

    The values this function returns are calculated using a dynamic
    program which avoids redundant computation across the subproblems
    in the solution. For array-like input, this implementation also
    avoids redundant computation across the different Stirling number
    calculations.

    The numbers are sometimes denoted

    .. math::

        {N \brace{K}}

    see [1]_ for details. This is often expressed-verbally-as
    "N subset K".

    Parameters
    ----------
    N : int, ndarray
        Number of things.
    K : int, ndarray
        Number of non-empty subsets taken.
    exact : bool, optional
        Uses dynamic programming (DP) with floating point
        numbers for smaller arrays and uses a second order approximation due to
        Temme for larger entries  of `N` and `K` that allows trading speed for
        accuracy. See [2]_ for a description. Temme approximation is used for
        values `n>50`. The max error from the DP has max relative error
        `4.5*10^-16` for `n<=50` and the max error from the Temme approximation
        has max relative error `5*10^-5` for `51 <= n < 70` and
        `9*10^-6` for `70 <= n < 101`. Note that these max relative errors will
        decrease further as `n` increases.

    Returns
    -------
    val : int, float, ndarray
        The number of partitions.

    See Also
    --------
    comb : The number of combinations of N things taken k at a time.

    Notes
    -----
    - If N < 0, or K < 0, then 0 is returned.
    - If K > N, then 0 is returned.

    The output type will always be `int` or ndarray of `object`.
    The input must contain either numpy or python integers otherwise a
    TypeError is raised.

    References
    ----------
    .. [1] R. L. Graham, D. E. Knuth and O. Patashnik, "Concrete
        Mathematics: A Foundation for Computer Science," Addison-Wesley
        Publishing Company, Boston, 1989. Chapter 6, page 258.

    .. [2] Temme, Nico M. "Asymptotic estimates of Stirling numbers."
        Studies in Applied Mathematics 89.3 (1993): 233-243.

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.special import stirling2
    >>> k = np.array([3, -1, 3])
    >>> n = np.array([10, 10, 9])
    >>> stirling2(n, k)
    array([9330, 0, 3025], dtype=object)

    """
    output_is_scalar = np.isscalar(N) and np.isscalar(K)
    # make a min-heap of unique (n,k) pairs
    N, K = asarray(N), asarray(K)
    if not np.issubdtype(N.dtype, np.integer):
        raise TypeError("Argument `N` must contain only integers")
    if not np.issubdtype(K.dtype, np.integer):
        raise TypeError("Argument `K` must contain only integers")
    if not exact:
        # NOTE: here we allow np.uint via casting to double types prior to
        # passing to private ufunc dispatcher. All dispatched functions
        # take double type for (n,k) arguments and return double.
        return _stirling2_inexact(N.astype(float), K.astype(float))
    nk_pairs = list(
        set([(n.take(0), k.take(0))
             for n, k in np.nditer([N, K], ['refs_ok'])])
    )
    heapify(nk_pairs)
    # base mapping for small values
    snsk_vals = defaultdict(int)
    for pair in [(0, 0), (1, 1), (2, 1), (2, 2)]:
        snsk_vals[pair] = 1
    # for each pair in the min-heap, calculate the value, store for later
    n_old, n_row = 2, [0, 1, 1]
    while nk_pairs:
        n, k = heappop(nk_pairs)
        if n < 2 or k > n or k <= 0:
            continue
        elif k == n or k == 1:
            snsk_vals[(n, k)] = 1
            continue
        elif n != n_old:
            num_iters = n - n_old
            while num_iters > 0:
                n_row.append(1)
                # traverse from back to remove second row
                for j in range(len(n_row)-2, 1, -1):
                    n_row[j] = n_row[j]*j + n_row[j-1]
                num_iters -= 1
            snsk_vals[(n, k)] = n_row[k]
        else:
            snsk_vals[(n, k)] = n_row[k]
        n_old, n_row = n, n_row
    out_types = [object, object, object] if exact else [float, float, float]
    # for each pair in the map, fetch the value, and populate the array
    it = np.nditer(
        [N, K, None],
        ['buffered', 'refs_ok'],
        [['readonly'], ['readonly'], ['writeonly', 'allocate']],
        op_dtypes=out_types,
    )
    with it:
        while not it.finished:
            it[2] = snsk_vals[(int(it[0]), int(it[1]))]
            it.iternext()
        output = it.operands[2]
        # If N and K were both scalars, convert output to scalar.
        if output_is_scalar:
            output = output.take(0)
    return output


def zeta(x, q=None, out=None):
    r"""
    Riemann or Hurwitz zeta function.

    Parameters
    ----------
    x : array_like of float
        Input data, must be real
    q : array_like of float, optional
        Input data, must be real.  Defaults to Riemann zeta.
    out : ndarray, optional
        Output array for the computed values.

    Returns
    -------
    out : array_like
        Values of zeta(x).

    See Also
    --------
    zetac

    Notes
    -----
    The two-argument version is the Hurwitz zeta function

    .. math::

        \zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x};

    see [dlmf]_ for details. The Riemann zeta function corresponds to
    the case when ``q = 1``.

    References
    ----------
    .. [dlmf] NIST, Digital Library of Mathematical Functions,
        https://dlmf.nist.gov/25.11#i

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.special import zeta, polygamma, factorial

    Some specific values:

    >>> zeta(2), np.pi**2/6
    (1.6449340668482266, 1.6449340668482264)

    >>> zeta(4), np.pi**4/90
    (1.0823232337111381, 1.082323233711138)

    Relation to the `polygamma` function:

    >>> m = 3
    >>> x = 1.25
    >>> polygamma(m, x)
    array(2.782144009188397)
    >>> (-1)**(m+1) * factorial(m) * zeta(m+1, x)
    2.7821440091883969

    """
    if q is None:
        return _ufuncs._riemann_zeta(x, out)
    else:
        return _ufuncs._zeta(x, q, out)