This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.


The (generalized) Laguerre polynomials

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.


orthpoly::laguerre(n, a, x)


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.


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)



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


An arithmetical expression.


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.


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 .

Was this topic helpful?