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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
..................
.. autofunction:: mpmath.legendre
.. autofunction:: mpmath.legenp
.. autofunction:: mpmath.legenq
Chebyshev polynomials
.....................
.. autofunction:: mpmath.chebyt
.. autofunction:: mpmath.chebyu
Jacobi polynomials
..................
.. autofunction:: mpmath.jacobi
Gegenbauer polynomials
......................
.. autofunction:: mpmath.gegenbauer
Hermite polynomials
...................
.. autofunction:: mpmath.hermite
Laguerre polynomials
....................
.. autofunction:: mpmath.laguerre
Spherical harmonics
...................
.. autofunction:: mpmath.spherharm
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