**gegenbauer_polynomial**,
a Python code which
evaluates Gegenbauer polynomials and associated functions.

The Gegenbauer polynomial C(n,alpha,x) can be defined by:

C(0,alpha,x) = 1 C(1,alpha,x) = 2 * alpha * x C(n,alpha,x) = (1/n) * ( 2*x*(n+alpha-1) * C(n-1,alpha,x) - (n+2*alpha-2) * C(n-2,alpha,x) )where n is a nonnegative integer, and -1/2 < alpha, 0 =/= alpha.

The N zeroes of C(n,alpha,x) are the abscissas used for Gauss-Gegenbauer quadrature of the integral of a function F(X) with weight function (1-x^2)^(alpha-1/2) over the interval [-1,1].

The Gegenbauer polynomials are orthogonal under the inner product defined as weighted integration from -1 to 1:

Integral ( -1 <= x <= 1 ) (1-x^2)^(alpha-1/2) * C(i,alpha,x) * C(j,alpha,x) dx = 0 if i =/= j = pi * 2^(1-2*alpha) * Gamma(n+2*alpha) / n! / (n+alpha) / (Gamma(alpha))^2 if i = j.

The information on this web page is distributed under the MIT license.

**gegenbauer_polynomial** is available in
a C version and
a C++ version and
a Fortran90 version and
a MATLAB version and
an Octave version and
a Python version.

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- gegenbauer_polynomial.py, the source code.
- gegenbauer_polynomial.sh, runs all the tests.
- gegenbauer_polynomial.txt, the output file.