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# orthpoly::laguerre

The (generalized) Laguerre polynomials

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## 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

Polynomials of domain type DOM_POLY are returned, if identifiers or indexed identifiers are specified:

`orthpoly::laguerre(2, a, x)`

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

However, using arithmetical expressions as input the "values" of these polynomials are returned:

`orthpoly::laguerre(2, 4, 6*x)`

`orthpoly::laguerre(2, 2/3*I, 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

If x is an indeterminate, then a polynomial of domain type DOM_POLY is returned. If x is an arithmetical expression, then the value of the Laguerre polynomial at this point 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 .