# GEGENBAUER_POLYNOMIAL Gegenbauer Polynomials

GEGENBAUER_POLYNOMIAL is a C++ 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 C++ library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

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

GEGENBAUER_RULE, a C++ program which can generate a Gauss-Gegenbauer quadrature rule on request.

HERMITE_POLYNOMIAL, a C++ library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

JACOBI_POLYNOMIAL, a C++ library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a C++ library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a C++ library which evaluates the Legendre polynomial and associated functions.

LEGENDRE_SHIFTED_POLYNOMIAL, a C++ library which evaluates the shifted Legendre polynomial, with domain [0,1].

LOBATTO_POLYNOMIAL, a C++ library which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

POLPAK, a C++ library which evaluates a variety of mathematical functions.

### List of Routines:

• GEGENBAUER_ALPHA_CHECK checks the value of ALPHA.
• GEGENBAUER_EK_COMPUTE computes a Gauss-Gegenbauer quadrature rule.
• GEGENBAUER_INTEGRAL: the integral of a monomial with Gegenbauer weight.
• GEGENBAUER_POLYNOMIAL_VALUE computes the Gegenbauer polynomials C(I,ALPHA)(X).
• GEGENBAUER_POLYNOMIAL_VALUES returns some values of the Gegenbauer polynomials.
• GEGENBAUER_SS_COMPUTE computes a Gauss-Gegenbauer quadrature rule.
• GEGENBAUER_SS_RECUR: value and derivative of a Gegenbauer polynomial.
• GEGENBAUER_SS_ROOT improves an approximate root of a Gegenbauer polynomial.
• HYPER_2F1_VALUES returns some values of the hypergeometric function 2F1.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• R8_EPSILON returns the R8 round off unit.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_HYPER_2F1 evaluates the hypergeometric function 2F1(A,B,C,X).
• R8_PSI evaluates the function Psi(X).
• R8_SIGN returns the sign of an R8.
• R8_UNIFORM_AB returns a pseudorandom R8 scaled to [A,B].
• R8VEC_PRINT prints an R8VEC.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.

Last revised on 30 November 2015.