laguerreL

Laguerre polynomials and L function

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

laguerreL(n, x)
laguerreL(n, a, x)

Description

laguerreL(n, a, x) represents Laguerre's L function. When n is a nonnegative integer, this is the classical Laguerre polynomial of degree n.

Laguerre's L function is defined in terms of hypergeometric functions by

.

For nonnegative integer values of n, the function returns the classical (generalized) polynomials that are orthogonal with respect to the scalar product . In particular:

.

The Laguerre's L function is not well defined for all values of the parameters n and a, because certain restrictions on the parameters exist in the definition of the hypergeometric functions . If the Laguerre's L function is not defined for a particular pair n and a, the call laguerreL(n, a, x) returns 0 or issues an error message.

The calls laguerre(n, x) and laguerre(n, 0, x) are equivalent.

If n is a nonnegative integer, the function laguerreL returns the explicit form of the corresponding Laguerre polynomial. The special values are implemented for arbitrary values of n and a. If n is a negative integer and a is a numerical noninteger value satisfying a ≥ - n, then the function laguerreL returns 0. If n is a negative integer and a is an integer satisfying a < - n, then the function returns an explicit expression defined by the reflection rule

.

If all arguments are numerical and at least one of the arguments is a floating-point number, then laguerreL(x) returns a floating-point number. For all other arguments, laguerreL(n, a, x) returns a symbolic function call.

Environment Interactions

When called with floating-point arguments, the function is sensitive to the environment variable DIGITS, which determines the numerical working precision.

Examples

Example 1

You can call the laguerreL function with exact and symbolic arguments:

laguerreL(2, a, x), laguerreL(-2, -2, PI)

If the first argument is a nonnegative integer, the function returns a polynomial:

laguerreL(3, x)

laguerreL(3, a, x)

Floating-point values are computed for floating-point arguments:

laguerreL(2, 3, 4.0), laguerreL(5.0, sqrt(2), PI)

laguerreL(1 + I, 1.0), laguerreL(-2.0, exp(I))

Example 2

The Laguerre function is not defined for all parameter values:

laguerreL(-5/2, -3/2, x)
Error: The function 'laguerreL' is not defined for parameter values '-5/2' and '-3/2'. [laguerreL]

Example 3

System functions such as diff, float, limit, and series handle expressions involving laguerreL:

diff(laguerreL(n, a, x), x, x, x), float(laguerreL(2, 3, sqrt(PI)))

limit(laguerreL(3, 4, x^2/(1+x)), x = infinity)

limit(laguerreL(4, 3, x^2/(1+x)), x = infinity)

series(laguerreL(n, a, x), x = 0, 3)

series(laguerreL(3/2, x), x = infinity, 3)

Parameters

n, a, x

arithmetical expressions

Return Values

Arithmetical expression.

Overloaded By

x

See Also

MuPAD Functions

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