# laguerre_polynomial

laguerre_polynomial, a FORTRAN90 code which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

The Laguerre polynomial L(n,x) can be defined by:

```        L(n,x) = exp(x)/n! * d^n/dx^n ( exp(-x) * x^n )
```
where n is a nonnegative integer.

The generalized Laguerre polynomial Lm(n,m,x) can be defined by:

```        Lm(n,m,x) = exp(x)/(x^m*n!) * d^n/dx^n ( exp(-x) * x^(m+n) )
```
where n and m are nonnegative integers.

The Laguerre function can be defined by:

```        Lf(n,alpha,x) = exp(x)/(x^alpha*n!) * d^n/dx^n ( exp(-x) * x^(alpha+n) )
```
where n is a nonnegative integer and -1.0 < alpha is a real number.

### Languages:

laguerre_polynomial is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a FORTRAN90 code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a FORTRAN90 code 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_POLYNOMIAL, a FORTRAN90 code which evaluates the Gegenbauer polynomial and associated functions.

GEN_LAGUERRE_RULE, a FORTRAN90 code which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_POLYNOMIAL, a FORTRAN90 code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

JACOBI_POLYNOMIAL, a FORTRAN90 code which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_EXACTNESS, a FORTRAN90 code which tests the polynomial exactness of Gauss-Laguerre quadrature rules.

LAGUERRE_RULE, a FORTRAN90 code which can compute and print a Gauss-Laguerre quadrature rule.

laguerre_integrands, a FORTRAN90 library which defines test integrands for Gauss-Laguerre quadrature over [A,+oo).

LEGENDRE_POLYNOMIAL, a FORTRAN90 code which evaluates the Legendre polynomial and associated functions.

LOBATTO_POLYNOMIAL, a FORTRAN90 code which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

POLPAK, a FORTRAN90 code which evaluates a variety of mathematical functions.

TEST_VALUES, a FORTRAN90 code which supplies test values of various mathematical functions.

### Reference:

1. Theodore Chihara,
An Introduction to Orthogonal Polynomials,
Gordon and Breach, 1978,
ISBN: 0677041500,
LC: QA404.5 C44.
2. Walter Gautschi,
Orthogonal Polynomials: Computation and Approximation,
Oxford, 2004,
ISBN: 0-19-850672-4,
LC: QA404.5 G3555.
3. Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
NIST Handbook of Mathematical Functions,
Cambridge University Press, 2010,
ISBN: 978-0521192255,
LC: QA331.N57.
4. Gabor Szego,
Orthogonal Polynomials,
American Mathematical Society, 1992,
ISBN: 0821810235,
LC: QA3.A5.v23.

### Source Code:

Last revised on 24 July 2020.