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Orthogonal polynomials
----------------------
An orthogonal polynomial sequence is a sequence of polynomials `P_0(x), P_1(x), \ldots` of degree `0, 1, \ldots`, which are mutually orthogonal in the sense that
.. math ::
\int_S P_n(x) P_m(x) w(x) dx =
\begin{cases}
c_n \ne 0 & \text{if $m = n$} \\
0 & \text{if $m \ne n$}
\end{cases}
where `S` is some domain (e.g. an interval `[a,b] \in \mathbb{R}`) and `w(x)` is a fixed *weight function*. A sequence of orthogonal polynomials is determined completely by `w`, `S`, and a normalization convention (e.g. `c_n = 1`). Applications of orthogonal polynomials include function approximation and solution of differential equations.
Orthogonal polynomials are sometimes defined using the differential equations they satisfy (as functions of `x`) or the recurrence relations they satisfy with respect to the order `n`. Other ways of defining orthogonal polynomials include differentiation formulas and generating functions. The standard orthogonal polynomials can also be represented as hypergeometric series (see :doc:`hypergeometric`), more specifically using the Gauss hypergeometric function `\,_2F_1` in most cases. The following functions are generally implemented using hypergeometric functions since this is computationally efficient and easily generalizes.
For more information, see the `Wikipedia article on orthogonal polynomials <http://en.wikipedia.org/wiki/Orthogonal_polynomials>`_.
Legendre functions
.......................................
:func:`legendre`
^^^^^^^^^^^^^^^^
.. autofunction:: mpmath.legendre(n, x)
:func:`legenp`
^^^^^^^^^^^^^^^
.. autofunction:: mpmath.legenp(n, m, z, type=2)
:func:`legenq`
^^^^^^^^^^^^^^^
.. autofunction:: mpmath.legenq(n, m, z, type=2)
Chebyshev polynomials
.....................
:func:`chebyt`
^^^^^^^^^^^^^^^
.. autofunction:: mpmath.chebyt(n, x)
:func:`chebyu`
^^^^^^^^^^^^^^^
.. autofunction:: mpmath.chebyu(n, x)
Jacobi polynomials
..................
:func:`jacobi`
^^^^^^^^^^^^^^
.. autofunction:: mpmath.jacobi(n, a, b, z)
Gegenbauer polynomials
.....................................
:func:`gegenbauer`
^^^^^^^^^^^^^^^^^^
.. autofunction:: mpmath.gegenbauer(n, a, z)
Hermite polynomials
.....................................
:func:`hermite`
^^^^^^^^^^^^^^^
.. autofunction:: mpmath.hermite(n, z)
Laguerre polynomials
.......................................
:func:`laguerre`
^^^^^^^^^^^^^^^^
.. autofunction:: mpmath.laguerre(n, a, z)
Spherical harmonics
.....................................
:func:`spherharm`
^^^^^^^^^^^^^^^^^
.. autofunction:: mpmath.spherharm(l, m, theta, phi)
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