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	#
	# 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(-8np.pi, 8np.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(nsubxsub)/(nsubdsub))
	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(vpii)K(v, z), phase = 1
	# For K, you can pull out the exp((v-k)pii) 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 += pL(v-n + i2, 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 + 56.1sqrt(q) - 134.7q + 90.7sqrt(q)q
	else:
	qm = 17.0 + 3.1sqrt(q) - .126q + .0037sqrt(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 + 56.1sqrt(q) - 134.7q + 90.7sqrt(q)q
	else:
	qm = 17.0 + 3.1sqrt(q) - .126q + .0037sqrt(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) = ((23)(45))((67)(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)