The (generalized) Laguerre polynomials
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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.
Polynomial expressions are returned if identifiers or indexed identifiers are specified:
orthpoly::laguerre(2, a, x)
orthpoly::laguerre(3, a, x)
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 + 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,
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
The value of the Laguerre polynomial at point
returned as an arithmetical expression. If
an arithmetical expression, then
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 .