# GEGENBAUER_POLYNOMIAL Gegenbauer Polynomials

GEGENBAUER_POLYNOMIAL is a Python library 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.
```

### Languages:

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

### Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a Python library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a Python library which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

GEGENBAUER_CC, a Python library which estimates the Gegenbauer weighted integral of a function f(x) using a Clenshaw-Curtis approach.

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### Examples and Tests:

You can go up one level to the Python source codes.

Last revised on 24 November 2015.