gegenbauer_polynomial, an Octave 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.

Related Data and Programs:


companion_matrix, an Octave code which computes the companion matrix for a polynomial. The polynomial may be represented in the standard monomial basis, or as a sum of Chebyshev, Gegenbauer, Hermite, Laguerre, or Lagrange basis polynomials. All the roots of the polynomial can be determined as the eigenvalues of the corresponding companion matrix.

gegenbauer_cc, an Octave code which estimates the Gegenbauer weighted integral of a function f(x) using a Clenshaw-Curtis approach.

gegenbauer_exactness, an Octave code which tests the monomial exactness of Gauss-Gegenbauer quadrature rules.

gegenbauer_rule, an Octave code which computes a Gauss-Gegenbauer quadrature rule.

octave_polynomial, an Octave code which analyzes a variety of polynomial families, returning the polynomial values, coefficients, derivatives, integrals, roots, or other information.

polpak, an Octave code which evaluates a variety of mathematical functions.

Source Code:

Last revised on 24 April 2024.