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 `_. 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