# Documentation

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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 .