**gegenbauer_polynomial**,
a MATLAB 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 computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

**gegenbauer_polynomial** is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version and
a Python version..

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- gegenbauer_alpha_check.m, checks that the Gegenbauer parameter ALPHA is legal.
- gegenbauer_ek_compute.m, computes a Gegenbauer quadrature rule using the Elhay-Kautsky algorithm.
- gegenbauer_integral.m, evaluates the integral of a monomial with Gegenbauer weight.
- gegenbauer_polynomial_value.m, evaluates the Gegenbauer polynomials C(n,alpha,x).
- gegenbauer_polynomial_values.m, returns selected values of the Gegenbauer polynomials C(n,alpha,x).
- gegenbauer_ss_compute.m, computes a Gegenbauer quadrature rule using the Stroud-Secrest algorithm.
- gegenbauer_ss_recur.m finds the value and derivative of a Gegenbauer polynomial.
- gegenbauer_ss_root.m improves an approximate root of a Gegenbauer polynomial.
- hyper_2f1_values.m, returns selected values of the hypergeometric function F(A,B,C,X).
- imtqlx.m diagonalizes a symmetric tridiagonal matrix.
- r8_hyper_2f1.m, evaluates the hypergeometric function F(A,B,C,X).
- r8_uniform_ab.m, returns a scaled pseudorandom R8;
- r8vec_print.m, prints an R8VEC;
- timestamp.m, prints the current YMDHMS date as a time stamp.