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orthpoly::laguerre

The (generalized) Laguerre polynomials

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Syntax

orthpoly::laguerre(n, a, x)

Description

orthpoly::laguerre(n,a,x) computes the value of the generalized n-th degree Laguerre polynomial with parameter a at the point x.

The standard Laguerre polynomials correspond to a = 0. They have rational coefficients.

Examples

Example 1

Polynomial expressions are returned if identifiers or indexed identifiers are specified:

orthpoly::laguerre(2, a, x)

orthpoly::laguerre(3, a, x[1])

Using arithmetical expressions as input, the “values” of these polynomials are returned:

orthpoly::laguerre(2, 4, 3+2*I)

orthpoly::laguerre(2, 2/3*I, exp(x[1] + 2))

“Arithmetical expressions” include numbers:

orthpoly::laguerre(2, a, sqrt(2)),
orthpoly::laguerre(3, 0.4, 8 + I),
orthpoly::laguerre(1000, 3, 0.3);

If the degree of the polynomial is a variable or expression, then orthpoly::laguerre returns itself symbolically:

orthpoly::laguerre(n, a, x)

Parameters

n

A nonnegative integer or an arithmetical expression representing a nonnegative integer: the degree of the polynomial.

a

An arithmetical expression.

x

An indeterminate or an arithmetical expression. An indeterminate is either an identifier (of domain type DOM_IDENT) or an indexed identifier (of type "_index").

Return Values

The value of the Laguerre polynomial at point x is returned as an arithmetical expression. If n is an arithmetical expression, then orthpoly::laguerre returns itself symbolically.

Algorithms

The Laguerre polynomials are given by the recursion formula

with L(0, a, x) = 1 and L(1, a, x) = 1 + a - x.

For fixed real a > - 1 these polynomials are orthogonal on the interval with respect to the weight function .

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