chore: import upstream snapshot with attribution

docs/functions/orthogonal.rst

@@ -0,0 +1,77 @@
+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