o
    8VaR                     @   s  d Z ddlmZ ddlmZmZ ddlmZ ddlm	Z	 ddl
mZmZmZ ddlmZ ddlmZ dd	lmZ dd
lmZ ddlmZmZ ddlmZ ddlmZ ddlmZmZm Z m!Z!m"Z"m#Z#m$Z$ e	dZ%G dd deZ&G dd de&Z'dd Z(G dd de&Z)G dd de&Z*G dd de&Z+G dd deZ,G dd deZ-G d d! d!e&Z.G d"d# d#eZ/G d$d% d%e&Z0G d&d' d'e&Z1G d(d) d)e&Z2d*S )+z
This module mainly implements special orthogonal polynomials.

See also functions.combinatorial.numbers which contains some
combinatorial polynomials.

    )Rational)FunctionArgumentIndexError)S)Dummy)binomial	factorialRisingFactorial)re)exp)floor)sqrt)cossec)gamma)hyper)jacobi_polygegenbauer_polychebyshevt_polychebyshevu_polylaguerre_polyhermite_polylegendre_polyxc                   @   s$   e Zd ZdZedd Zdd ZdS )OrthogonalPolynomialz+Base class for orthogonal polynomials.
    c                 C   s.   |j r|dkr| t|tt|S d S d S )Nr   )
is_integer_ortho_polyint_xsubsclsnr    r#   E/usr/lib/python3/dist-packages/sympy/functions/special/polynomials.py_eval_at_order'   s   z#OrthogonalPolynomial._eval_at_orderc                 C   s   |  | jd | jd  S )Nr      )funcargs	conjugate)selfr#   r#   r$   _eval_conjugate,   s   z$OrthogonalPolynomial._eval_conjugateN)__name__
__module____qualname____doc__classmethodr%   r+   r#   r#   r#   r$   r   #   s
    
r   c                   @   6   e Zd ZdZedd ZdddZdd Zd	d
 ZdS )jacobia  
    Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$.

    Explanation
    ===========

    ``jacobi(n, alpha, beta, x)`` gives the nth Jacobi polynomial
    in x, $P_n^{\left(\alpha, \beta\right)}(x)$.

    The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect
    to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$.

    Examples
    ========

    >>> from sympy import jacobi, S, conjugate, diff
    >>> from sympy.abc import a, b, n, x

    >>> jacobi(0, a, b, x)
    1
    >>> jacobi(1, a, b, x)
    a/2 - b/2 + x*(a/2 + b/2 + 1)
    >>> jacobi(2, a, b, x)
    a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2

    >>> jacobi(n, a, b, x)
    jacobi(n, a, b, x)

    >>> jacobi(n, a, a, x)
    RisingFactorial(a + 1, n)*gegenbauer(n,
        a + 1/2, x)/RisingFactorial(2*a + 1, n)

    >>> jacobi(n, 0, 0, x)
    legendre(n, x)

    >>> jacobi(n, S(1)/2, S(1)/2, x)
    RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)

    >>> jacobi(n, -S(1)/2, -S(1)/2, x)
    RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)

    >>> jacobi(n, a, b, -x)
    (-1)**n*jacobi(n, b, a, x)

    >>> jacobi(n, a, b, 0)
    gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(2**n*factorial(n)*gamma(a + 1))
    >>> jacobi(n, a, b, 1)
    RisingFactorial(a + 1, n)/factorial(n)

    >>> conjugate(jacobi(n, a, b, x))
    jacobi(n, conjugate(a), conjugate(b), conjugate(x))

    >>> diff(jacobi(n,a,b,x), x)
    (a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)

    See Also
    ========

    gegenbauer,
    chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly,
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
    .. [2] http://mathworld.wolfram.com/JacobiPolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/JacobiP/

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|| d t
|d  d| |d   d| |d   t|| | S |js| rtj| t||||  S |jrd|  t
|| d  t
|d t|  t| | | g|d gd S |tjkrt|d |t| S |tju r|jr|| d|  jrtdt|| | d |tj S d S d S t||||S )N      r&   z,Error. a + b + 2*n should not be an integer.)r   r	   r   Halfr   
chebyshevtis_zerolegendre
chebyshevu
gegenbauerr   assoc_legendre	is_Numbercould_extract_minus_signNegativeOner2   r   OneInfinityis_positiver   
ValueErrorr   )r!   r"   abr   r#   r#   r$   eval   s6   

&2
J,

zjacobi.eval   c           
   	   C   s  ddl m} |dkrt| ||dkrj| j\}}}}td}d|| | | d  }|| d|  d t|| d ||  || t|| | d ||   }	||t|||||	t||||   |d|d fS |dkr| j\}}}}td}d|| | | d  }d||  || d|  d t|| d ||  || t|| | d ||    }	||t|||||	t||||   |d|d fS |dkr| j\}}}}tj|| | d  t|d |d |d | S t| |)	Nr   Sumr&   r4   kr5   r3   rG   )	sympyrI   r   r(   r   r	   r2   r   r6   )
r*   argindexrI   r"   rD   rE   r   rJ   f1f2r#   r#   r$   fdiff   s.   
( 42 40
zjacobi.fdiffc           	      K   s   ddl m} |js|jdu rtdtd}t| |t|| | d | t|| d ||  t| d| d |  }dt| |||d|f S )Nr   rH   F*Error: n should be a non-negative integer.rJ   r&   r4   )rK   rI   is_negativer   rC   r   r	   r   )	r*   r"   rD   rE   r   kwargsrI   rJ   kernr#   r#   r$   _eval_rewrite_as_polynomial   s   6z"jacobi._eval_rewrite_as_polynomialc                 C   s*   | j \}}}}| || | | S Nr(   r'   r)   )r*   r"   rD   rE   r   r#   r#   r$   r+      s   zjacobi._eval_conjugateN)rG   	r,   r-   r.   r/   r0   rF   rO   rT   r+   r#   r#   r#   r$   r2   4   s    P

%
r2   c                 C   sz   t d|| d  t| | d t| | d   d|  | | d  t| t| | | d   }t| |||t| S )a  
    Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$.

    Explanation
    ===========

    ``jacobi_normalized(n, alpha, beta, x)`` gives the nth
    Jacobi polynomial in *x*, $P_n^{\left(\alpha, \beta\right)}(x)$.

    The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect
    to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$.

    This functions returns the polynomials normilzed:

    .. math::

        \int_{-1}^{1}
          P_m^{\left(\alpha, \beta\right)}(x)
          P_n^{\left(\alpha, \beta\right)}(x)
          (1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x
        = \delta_{m,n}

    Examples
    ========

    >>> from sympy import jacobi_normalized
    >>> from sympy.abc import n,a,b,x

    >>> jacobi_normalized(n, a, b, x)
    jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1)))

    Parameters
    ==========

    n : integer degree of polynomial

    a : alpha value

    b : beta value

    x : symbol

    See Also
    ========

    gegenbauer,
    chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly,
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
    .. [2] http://mathworld.wolfram.com/JacobiPolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/JacobiP/

    r4   r&   )r   r   r   r2   r   )r"   rD   rE   r   Znfactorr#   r#   r$   jacobi_normalized   s   2CrX   c                   @   r1   )r;   a  
    Gegenbauer polynomial $C_n^{\left(\alpha\right)}(x)$.

    Explanation
    ===========

    ``gegenbauer(n, alpha, x)`` gives the nth Gegenbauer polynomial
    in x, $C_n^{\left(\alpha\right)}(x)$.

    The Gegenbauer polynomials are orthogonal on $[-1, 1]$ with
    respect to the weight $\left(1-x^2\right)^{\alpha-\frac{1}{2}}$.

    Examples
    ========

    >>> from sympy import gegenbauer, conjugate, diff
    >>> from sympy.abc import n,a,x
    >>> gegenbauer(0, a, x)
    1
    >>> gegenbauer(1, a, x)
    2*a*x
    >>> gegenbauer(2, a, x)
    -a + x**2*(2*a**2 + 2*a)
    >>> gegenbauer(3, a, x)
    x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)

    >>> gegenbauer(n, a, x)
    gegenbauer(n, a, x)
    >>> gegenbauer(n, a, -x)
    (-1)**n*gegenbauer(n, a, x)

    >>> gegenbauer(n, a, 0)
    2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1))
    >>> gegenbauer(n, a, 1)
    gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))

    >>> conjugate(gegenbauer(n, a, x))
    gegenbauer(n, conjugate(a), conjugate(x))

    >>> diff(gegenbauer(n, a, x), x)
    2*a*gegenbauer(n - 1, a + 1, x)

    See Also
    ========

    jacobi,
    chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Gegenbauer_polynomials
    .. [2] http://mathworld.wolfram.com/GegenbauerPolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/GegenbauerC3/

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S ttj||  ttj|  td| |  td| t|d   S | rjtj| t|||  S |jrd| ttj t|tj|   td| d t|d  t|  S |tjkrtd| | td| t|d   S |tju r|jrt||tj S d S d S t|||S )NTr4   r&   )rQ   r   Zeror6   r9   r@   r:   r?   r=   r
   ZComplexInfinityr   Pir   r   r>   r;   r8   r   rA   rB   r	   r   )r!   r"   rD   r   r#   r#   r$   rF   h  s:   

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
.""
(
zgegenbauer.evalr5   c           
      C   s  ddl m} |dkrt| ||dkrq| j\}}}td}ddd||    ||  || d|  ||   }d|d  |d|  d| d|  d   d|| d|    }|t||| |t|||  }	||	|d|d fS |dkr| j\}}}d| t|d |d | S t| |)Nr   rH   r&   r4   rJ   r3   r5   )rK   rI   r   r(   r   r;   )
r*   rL   rI   r"   rD   r   rJ   Zfactor1Zfactor2rS   r#   r#   r$   rO     s,   
* 
zgegenbauer.fdiffc                 K   sn   ddl m} td}d| t|||  d| |d|    t|t|d|    }|||dt|d fS Nr   rH   rJ   r3   r4   )rK   rI   r   r	   r   r   )r*   r"   rD   r   rR   rI   rJ   rS   r#   r#   r$   rT     s   (z&gegenbauer._eval_rewrite_as_polynomialc                 C   "   | j \}}}| || | S rU   rV   )r*   r"   rD   r   r#   r#   r$   r+        zgegenbauer._eval_conjugateNr5   rW   r#   r#   r#   r$   r;   $  s    C

(r;   c                   @   6   e Zd ZdZeeZedd Zd
ddZ	dd Z
d	S )r7   a  
    Chebyshev polynomial of the first kind, $T_n(x)$.

    Explanation
    ===========

    ``chebyshevt(n, x)`` gives the nth Chebyshev polynomial (of the first
    kind) in x, $T_n(x)$.

    The Chebyshev polynomials of the first kind are orthogonal on
    $[-1, 1]$ with respect to the weight $\frac{1}{\sqrt{1-x^2}}$.

    Examples
    ========

    >>> from sympy import chebyshevt, diff
    >>> from sympy.abc import n,x
    >>> chebyshevt(0, x)
    1
    >>> chebyshevt(1, x)
    x
    >>> chebyshevt(2, x)
    2*x**2 - 1

    >>> chebyshevt(n, x)
    chebyshevt(n, x)
    >>> chebyshevt(n, -x)
    (-1)**n*chebyshevt(n, x)
    >>> chebyshevt(-n, x)
    chebyshevt(n, x)

    >>> chebyshevt(n, 0)
    cos(pi*n/2)
    >>> chebyshevt(n, -1)
    (-1)**n

    >>> diff(chebyshevt(n, x), x)
    n*chebyshevu(n - 1, x)

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
    .. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
    .. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
    .. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/
    .. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/

    c                 C   s   |j s;| rtj| t||  S | rt| |S |jr)ttjtj | S |tj	kr1tj	S |tj
u r9tj
S d S |jrE| | |S | ||S rU   )r=   r>   r   r?   r7   r8   r   r6   rZ   r@   rA   rQ   r%   r    r#   r#   r$   rF     s   

zchebyshevt.evalr4   c                 C   s@   |dkr	t | ||dkr| j\}}|t|d | S t | |Nr&   r4   )r   r(   r:   r*   rL   r"   r   r#   r#   r$   rO        


zchebyshevt.fdiffc                 K   sZ   ddl m} td}t|d| |d d |  ||d|    }|||dt|d fS )Nr   rH   rJ   r4   r&   )rK   rI   r   r   r   r*   r"   r   rR   rI   rJ   rS   r#   r#   r$   rT      s   .z&chebyshevt._eval_rewrite_as_polynomialNr4   )r,   r-   r.   r/   staticmethodr   r   r0   rF   rO   rT   r#   r#   r#   r$   r7     s    B

r7   c                   @   r_   )r:   a  
    Chebyshev polynomial of the second kind, $U_n(x)$.

    Explanation
    ===========

    ``chebyshevu(n, x)`` gives the nth Chebyshev polynomial of the second
    kind in x, $U_n(x)$.

    The Chebyshev polynomials of the second kind are orthogonal on
    $[-1, 1]$ with respect to the weight $\sqrt{1-x^2}$.

    Examples
    ========

    >>> from sympy import chebyshevu, diff
    >>> from sympy.abc import n,x
    >>> chebyshevu(0, x)
    1
    >>> chebyshevu(1, x)
    2*x
    >>> chebyshevu(2, x)
    4*x**2 - 1

    >>> chebyshevu(n, x)
    chebyshevu(n, x)
    >>> chebyshevu(n, -x)
    (-1)**n*chebyshevu(n, x)
    >>> chebyshevu(-n, x)
    -chebyshevu(n - 2, x)

    >>> chebyshevu(n, 0)
    cos(pi*n/2)
    >>> chebyshevu(n, 1)
    n + 1

    >>> diff(chebyshevu(n, x), x)
    (-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
    .. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
    .. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
    .. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/
    .. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/

    c                 C   s   |j sO| rtj| t||  S | r.|tjkrtjS | d  s.t| d | S |jr;ttjtj	 | S |tj
krEtj
| S |tju rMtjS d S |jrd|tjkrZtjS | | d | S | ||S )Nr4   )r=   r>   r   r?   r:   rY   r8   r   r6   rZ   r@   rA   rQ   r%   r    r#   r#   r$   rF   l  s(   




zchebyshevu.evalr4   c                 C   s^   |dkr	t | ||dkr*| j\}}|d t|d | |t||  |d d  S t | |r`   )r   r(   r7   r:   ra   r#   r#   r$   rO     s   

0
zchebyshevu.fdiffc                 K   sn   ddl m} td}tj| t||  d| |d|    t|t|d|    }|||dt|d fS )Nr   rH   rJ   r4   )rK   rI   r   r   r?   r   r   rc   r#   r#   r$   rT     s   
z&chebyshevu._eval_rewrite_as_polynomialNrd   )r,   r-   r.   r/   re   r   r   r0   rF   rO   rT   r#   r#   r#   r$   r:   '  s    B

r:   c                   @      e Zd ZdZedd ZdS )chebyshevt_rootaK  
    ``chebyshev_root(n, k)`` returns the kth root (indexed from zero) of
    the nth Chebyshev polynomial of the first kind; that is, if
    0 <= k < n, ``chebyshevt(n, chebyshevt_root(n, k)) == 0``.

    Examples
    ========

    >>> from sympy import chebyshevt, chebyshevt_root
    >>> chebyshevt_root(3, 2)
    -sqrt(3)/2
    >>> chebyshevt(3, chebyshevt_root(3, 2))
    0

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly
    c                 C   s>   d|kr||k st d||f ttjd| d  d|  S )Nr   +must have 0 <= k < n, got k = %s and n = %sr4   r&   rC   r   r   rZ   r!   r"   rJ   r#   r#   r$   rF     s
   zchebyshevt_root.evalNr,   r-   r.   r/   r0   rF   r#   r#   r#   r$   rg         rg   c                   @   rf   )chebyshevu_roota5  
    ``chebyshevu_root(n, k)`` returns the kth root (indexed from zero) of the
    nth Chebyshev polynomial of the second kind; that is, if 0 <= k < n,
    ``chebyshevu(n, chebyshevu_root(n, k)) == 0``.

    Examples
    ========

    >>> from sympy import chebyshevu, chebyshevu_root
    >>> chebyshevu_root(3, 2)
    -sqrt(2)/2
    >>> chebyshevu(3, chebyshevu_root(3, 2))
    0

    See Also
    ========

    chebyshevt, chebyshevt_root, chebyshevu,
    legendre, assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly
    c                 C   s:   d|kr||k st d||f ttj|d  |d  S )Nr   rh   r&   ri   rj   r#   r#   r$   rF     s
   zchebyshevu_root.evalNrk   r#   r#   r#   r$   rm     rl   rm   c                   @   r_   )r9   am  
    ``legendre(n, x)`` gives the nth Legendre polynomial of x, $P_n(x)$

    Explanation
    ===========

    The Legendre polynomials are orthogonal on [-1, 1] with respect to
    the constant weight 1. They satisfy $P_n(1) = 1$ for all n; further,
    $P_n$ is odd for odd n and even for even n.

    Examples
    ========

    >>> from sympy import legendre, diff
    >>> from sympy.abc import x, n
    >>> legendre(0, x)
    1
    >>> legendre(1, x)
    x
    >>> legendre(2, x)
    3*x**2/2 - 1/2
    >>> legendre(n, x)
    legendre(n, x)
    >>> diff(legendre(n,x), x)
    n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
    assoc_legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Legendre_polynomial
    .. [2] http://mathworld.wolfram.com/LegendrePolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/LegendreP/
    .. [4] http://functions.wolfram.com/Polynomials/LegendreP2/

    c                 C   s   |j sR| rtj| t||  S | r&| d  s&t| tj |S |jr@ttjt	tj
|d  t	tj|d    S |tjkrHtjS |tju rPtjS d S |jr[| tj }| ||S r`   )r=   r>   r   r?   r9   r@   r8   r   rZ   r   r6   rA   rQ   r%   r    r#   r#   r$   rF   *  s   .

zlegendre.evalr4   c                 C   sZ   |dkr	t | ||dkr(| j\}}||d d  |t|| t|d |  S t | |r`   )r   r(   r9   ra   r#   r#   r$   rO   B  s   

,
zlegendre.fdiffc                 K   s^   ddl m} td}d| t||d  d| d ||   d| d |  }|||d|fS )Nr   rH   rJ   r3   r4   r&   )rK   rI   r   r   rc   r#   r#   r$   rT   Z  s   :z$legendre._eval_rewrite_as_polynomialNrd   )r,   r-   r.   r/   re   r   r   r0   rF   rO   rT   r#   r#   r#   r$   r9     s    4

r9   c                   @   sB   e Zd ZdZedd Zedd ZdddZd	d
 Zdd Z	dS )r<   aH  
    ``assoc_legendre(n, m, x)`` gives $P_n^m(x)$, where n and m are
    the degree and order or an expression which is related to the nth
    order Legendre polynomial, $P_n(x)$ in the following manner:

    .. math::
        P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}}
                   \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}

    Explanation
    ===========

    Associated Legendre polynomials are orthogonal on [-1, 1] with:

    - weight = 1            for the same m, and different n.
    - weight = 1/(1-x**2)   for the same n, and different m.

    Examples
    ========

    >>> from sympy import assoc_legendre
    >>> from sympy.abc import x, m, n
    >>> assoc_legendre(0,0, x)
    1
    >>> assoc_legendre(1,0, x)
    x
    >>> assoc_legendre(1,1, x)
    -sqrt(1 - x**2)
    >>> assoc_legendre(n,m,x)
    assoc_legendre(n, m, x)

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre,
    hermite,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
    .. [2] http://mathworld.wolfram.com/LegendrePolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/LegendreP/
    .. [4] http://functions.wolfram.com/Polynomials/LegendreP2/

    c                 C   s>   t |tddt|f}d| dtd  t|d  |  S )NT)Zpolysr3   r&   r4   )r   r   Zdiffr   Zas_expr)r!   r"   mPr#   r#   r$   r%     s   &zassoc_legendre._eval_at_orderc                 C   s
  |  rtj|  t|| t||   t|| | S |dkr&t||S |dkrGd| ttj td| | d td|| d    S |j	r}|j	r|j
r|j
r|jr^td| |f t||krmtd| ||f | t|tt|t|S d S d S d S d S )Nr   r4   r&   z3%s : 1st index must be nonnegative integer (got %r)z9%s : abs('2nd index') must be <= '1st index' (got %r, %r))r>   r   r?   r   r<   r9   r   rZ   r   r=   r   rQ   rC   absr%   r   r   r   )r!   r"   rn   r   r#   r#   r$   rF     s   2
: zassoc_legendre.evalr5   c                 C   s~   |dkr	t | ||dkrt | ||dkr:| j\}}}d|d d  || t||| || t|d ||   S t | |)Nr&   r4   r5   )r   r(   r<   )r*   rL   r"   rn   r   r#   r#   r$   rO     s   

<
zassoc_legendre.fdiffc                 K   s   ddl m} td}td| d|  d| t||  t| t|d|  |   d|  ||| d|    }d|d  |d  |||dt|| tj f S )Nr   rH   rJ   r4   r3   r&   )rK   rI   r   r   r   r   r6   )r*   r"   rn   r   rR   rI   rJ   rS   r#   r#   r$   rT     s   &2z*assoc_legendre._eval_rewrite_as_polynomialc                 C   r\   rU   rV   )r*   r"   rn   r   r#   r#   r$   r+     r]   zassoc_legendre._eval_conjugateNr^   )
r,   r-   r.   r/   r0   r%   rF   rO   rT   r+   r#   r#   r#   r$   r<   a  s    9


r<   c                   @   r_   )hermitea  
    ``hermite(n, x)`` gives the nth Hermite polynomial in x, $H_n(x)$

    Explanation
    ===========

    The Hermite polynomials are orthogonal on $(-\infty, \infty)$
    with respect to the weight $\exp\left(-x^2\right)$.

    Examples
    ========

    >>> from sympy import hermite, diff
    >>> from sympy.abc import x, n
    >>> hermite(0, x)
    1
    >>> hermite(1, x)
    2*x
    >>> hermite(2, x)
    4*x**2 - 2
    >>> hermite(n, x)
    hermite(n, x)
    >>> diff(hermite(n,x), x)
    2*n*hermite(n - 1, x)
    >>> hermite(n, -x)
    (-1)**n*hermite(n, x)

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    laguerre, assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Hermite_polynomial
    .. [2] http://mathworld.wolfram.com/HermitePolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/HermiteH/

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u r/tj
S d S |jr:td| | ||S )Nr4   0The index n must be nonnegative integer (got %r))r=   r>   r   r?   rq   r8   r   rZ   r   r@   rA   rQ   rC   r%   r    r#   r#   r$   rF     s   $
zhermite.evalr4   c                 C   sD   |dkr	t | ||dkr| j\}}d| t|d | S t | |r`   )r   r(   rq   ra   r#   r#   r$   rO     s   


zhermite.fdiffc                 K   sh   ddl m} td}d| t|t|d|    d| |d|    }t||||dt|d f S r[   )rK   rI   r   r   r   rc   r#   r#   r$   rT   %  s   4 z#hermite._eval_rewrite_as_polynomialNrd   )r,   r-   r.   r/   re   r   r   r0   rF   rO   rT   r#   r#   r#   r$   rq     s    3

rq   c                   @   r_   )laguerrea  
    Returns the nth Laguerre polynomial in x, $L_n(x)$.

    Examples
    ========

    >>> from sympy import laguerre, diff
    >>> from sympy.abc import x, n
    >>> laguerre(0, x)
    1
    >>> laguerre(1, x)
    1 - x
    >>> laguerre(2, x)
    x**2/2 - 2*x + 1
    >>> laguerre(3, x)
    -x**3/6 + 3*x**2/2 - 3*x + 1

    >>> laguerre(n, x)
    laguerre(n, x)

    >>> diff(laguerre(n, x), x)
    -assoc_laguerre(n - 1, 1, x)

    Parameters
    ==========

    n : int
        Degree of Laguerre polynomial. Must be ``n >= 0``.

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    assoc_laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial
    .. [2] http://mathworld.wolfram.com/LaguerrePolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/LaguerreL/
    .. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/

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S |tj
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 S d S |jrQt|t| d |  S | ||S )NFError: n should be an integer.r&   )r   rC   r=   r>   r   rs   r8   r   r@   NegativeInfinityrA   r?   rQ   r%   r    r#   r#   r$   rF   j  s   


zlaguerre.evalr4   c                 C   s@   |dkr	t | ||dkr| j\}}t|d d| S t | |r`   )r   r(   assoc_laguerrera   r#   r#   r$   rO     rb   zlaguerre.fdiffc                 K   s   ddl m} |jrt|| j| d | fi | S |jdu r$tdtd}t| |t	|d  ||  }|||d|fS )Nr   rH   r&   Frt   rJ   r4   )
rK   rI   rQ   r   rT   r   rC   r   r	   r   rc   r#   r#   r$   rT     s   $
 z$laguerre._eval_rewrite_as_polynomialNrd   )r,   r-   r.   r/   re   r   r   r0   rF   rO   rT   r#   r#   r#   r$   rs   0  s    7

rs   c                   @   r1   )rv   a)  
    Returns the nth generalized Laguerre polynomial in x, $L_n(x)$.

    Examples
    ========

    >>> from sympy import assoc_laguerre, diff
    >>> from sympy.abc import x, n, a
    >>> assoc_laguerre(0, a, x)
    1
    >>> assoc_laguerre(1, a, x)
    a - x + 1
    >>> assoc_laguerre(2, a, x)
    a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1
    >>> assoc_laguerre(3, a, x)
    a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) +
        x*(-a**2/2 - 5*a/2 - 3) + 1

    >>> assoc_laguerre(n, a, 0)
    binomial(a + n, a)

    >>> assoc_laguerre(n, a, x)
    assoc_laguerre(n, a, x)

    >>> assoc_laguerre(n, 0, x)
    laguerre(n, x)

    >>> diff(assoc_laguerre(n, a, x), x)
    -assoc_laguerre(n - 1, a + 1, x)

    >>> diff(assoc_laguerre(n, a, x), a)
    Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))

    Parameters
    ==========

    n : int
        Degree of Laguerre polynomial. Must be ``n >= 0``.

    alpha : Expr
        Arbitrary expression. For ``alpha=0`` regular Laguerre
        polynomials will be generated.

    See Also
    ========

    jacobi, gegenbauer,
    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
    legendre, assoc_legendre,
    hermite,
    laguerre,
    sympy.polys.orthopolys.jacobi_poly
    sympy.polys.orthopolys.gegenbauer_poly
    sympy.polys.orthopolys.chebyshevt_poly
    sympy.polys.orthopolys.chebyshevu_poly
    sympy.polys.orthopolys.hermite_poly
    sympy.polys.orthopolys.legendre_poly
    sympy.polys.orthopolys.laguerre_poly

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial#Generalized_Laguerre_polynomials
    .. [2] http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
    .. [3] http://functions.wolfram.com/Polynomials/LaguerreL/
    .. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/

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zassoc_laguerre.evalr5   c                 C   s   ddl m} |dkrt| ||dkr/| j\}}}td}|t|||||  |d|d fS |dkrD| j\}}}t|d |d | S t| |)Nr   rH   r&   r4   rJ   r5   )rK   rI   r   r(   r   rv   )r*   rL   rI   r"   rw   r   rJ   r#   r#   r$   rO     s   
$
zassoc_laguerre.fdiffc                 K   s   ddl m} |js|jdu rtdtd}t| |t|| d t|  ||  }t|| d t| |||d|f S )Nr   rH   FrP   rJ   r&   )	rK   rI   rQ   r   rC   r   r	   r   r   )r*   r"   rw   r   rR   rI   rJ   rS   r#   r#   r$   rT     s   (z*assoc_laguerre._eval_rewrite_as_polynomialc                 C   r\   rU   rV   )r*   r"   rw   r   r#   r#   r$   r+     r]   zassoc_laguerre._eval_conjugateNr^   rW   r#   r#   r#   r$   rv     s    E


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