# Documentation

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# `laguerreL`

Laguerre polynomials and L function

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## 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: Function 'laguerreL' not supported 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.

`x`