laguerre_polynomial


laguerre_polynomial, a Python 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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license.

Languages:

laguerre_polynomial is available in a C version and a C++ version and a Fortran77 version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

laguerre_rule, a Python code which computes and prints a Gauss-Laguerre quadrature rule.

polpak, a Python code which evaluates a variety of mathematical functions.

polynomial_conversion, a Python code which converts representations of a polynomial between monomial, Bernstein, Chebyshev, Hermite, Laguerre, Legendre, and other forms.

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

test_values, a Python 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 February 2024.