# laguerreL

Generalized Laguerre Function and Laguerre Polynomials

## Syntax

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

## Description

example

````laguerreL(n,x)` returns the Laguerre polynomial of degree `n` if `n` is a nonnegative integer. When `n` is not a nonnegative integer, `laguerreL` returns the Laguerre function. For details, see Generalized Laguerre Function.```

example

````laguerreL(n,a,x)` returns the generalized Laguerre polynomial of degree `n` if `n` is a nonnegative integer. When `n` is not a nonnegative integer, `laguerreL` returns the generalized Laguerre function.```

## Examples

### Find Laguerre Polynomials for Numeric and Symbolic Inputs

Find the Laguerre polynomial of degree `3` for input `4.3`.

```laguerreL(3,4.3) ```
```ans = 2.5838```

Find the Laguerre polynomial for symbolic inputs. Specify degree `n` as `3` to return the explicit form of the polynomial.

```syms x laguerreL(3,x) ```
```ans = - x^3/6 + (3*x^2)/2 - 3*x + 1```

If the degree of the Laguerre polynomial `n` is not specified, `laguerreL` cannot find the polynomial. When `laguerreL` cannot find the polynomial, it returns the function call.

```syms n x laguerreL(n,x) ```
```ans = laguerreL(n, x)```

### Find Generalized Laguerre Polynomial

Find the explicit form of the generalized Laguerre polynomial `L(n,a,x)` of degree `n = 2`.

```syms a x laguerreL(2,a,x)```
```ans = (3*a)/2 - x*(a + 2) + a^2/2 + x^2/2 + 1```

### Return Generalized Laguerre Function

When `n` is not a nonnegative integer, `laguerreL(n,a,x)` returns the generalized Laguerre function.

`laguerreL(-2.7,3,2)`
```ans = 0.2488```

`laguerreL` is not defined for certain inputs and returns an error.

```syms x laguerreL(-5/2, -3/2, x)```
```Error using symengine Function 'laguerreL' not supported for parameter values '-5/2' and '-3/2'.```

### Find Laguerre Polynomial with Vector and Matrix Inputs

Find the Laguerre polynomials of degrees `1` and `2` by setting `n = [1 2]`.

```syms x laguerreL([1 2],x)```
```ans = [ 1 - x, x^2/2 - 2*x + 1]```

`laguerreL` acts element-wise on `n` to return a vector with two elements.

If multiple inputs are specified as a vector, matrix, or multidimensional array, the inputs must be the same size. Find the generalized Laguerre polynomials where input arguments `n` and `x` are matrices.

```syms a n = [2 3; 1 2]; xM = [x^2 11/7; -3.2 -x]; laguerreL(n,a,xM)```
```ans = [ a^2/2 - a*x^2 + (3*a)/2 + x^4/2 - 2*x^2 + 1,... a^3/6 + (3*a^2)/14 - (253*a)/294 - 676/1029] [ a + 21/5,... a^2/2 + a*x + (3*a)/2 + x^2/2 + 2*x + 1]```

`laguerreL` acts element-wise on `n` and `x` to return a matrix of the same size as `n` and `x`.

### Differentiate and Find Limits of Laguerre Polynomials

Use `limit` to find the limit of a generalized Laguerre polynomial of degree `3` as `x` tends to ∞.

```syms x expr = laguerreL(3,2,x); limit(expr,x,Inf)```
```ans = -Inf```

Use `diff` to find the third derivative of the generalized Laguerre polynomial `laguerreL(n,a,x)`.

```syms n a expr = laguerreL(n,a,x); diff(expr,x,3)```
```ans = -laguerreL(n - 3, a + 3, x)```

### Find Taylor Series Expansion of Laguerre Polynomials

Use `taylor` to find the Taylor series expansion of the generalized Laguerre polynomial of degree `2` at `x = 0`.

```syms a x expr = laguerreL(2,a,x); taylor(expr,x)```
```ans = (3*a)/2 - x*(a + 2) + a^2/2 + x^2/2 + 1```

### Plot Laguerre Polynomials

Plot the Laguerre polynomials of orders `1` through `4`.

```syms x fplot(laguerreL(1:4,x)) axis([-2 10 -10 10]) grid on ylabel('L_n(x)') title('Laguerre polynomials of orders 1 through 4') legend('1','2','3','4','Location','best')```

## Input Arguments

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Degree of polynomial, specified as a number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, or multidimensional array.

Input, specified as a number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, or multidimensional array.

Input, specified as a number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, or multidimensional array.

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### Generalized Laguerre Function

The generalized Laguerre function is defined in terms of the hypergeometric function as

`$\text{laguerreL}\left(n,a,x\right)=\left(\begin{array}{c}n+a\\ a\end{array}\right){}_{1}F{}_{1}\left(-n;a+1;x\right).$`

For nonnegative integer values of `n`, the function returns the generalized Laguerre polynomials that are orthogonal with respect to the scalar product

`$〈f1,f2〉=\underset{0}{\overset{\infty }{\int }}{e}^{-x}{x}^{a}f1\left(x\right)f2\left(x\right)dx.$`

In particular, the generalized Laguerre polynomials satisfy this normalization.

## Algorithms

• The generalized Laguerre function is not defined for all values of parameters `n` and `a` because certain restrictions on the parameters exist in the definition of the hypergeometric functions. If the generalized Laguerre function is not defined for a particular pair of `n` and `a`, the `laguerreL` function returns an error message. See Return Generalized Laguerre Function.

• The calls `laguerreL(n,x)` and `laguerreL(n,0,x)` are equivalent.

• If `n` is a nonnegative integer, the `laguerreL` function returns the explicit form of the corresponding Laguerre polynomial.

• The special values $\mathrm{laguerreL}\left(n,a,0\right)=\left(\begin{array}{c}n+a\\ a\end{array}\right)$ 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 `laguerreL` returns `0`.

• If `n` is a negative integer and `a` is an integer satisfying a < -n, the function returns an explicit expression defined by the reflection rule

`$\mathrm{laguerreL}\left(n,a,x\right)={\left(-1\right)}^{a}{e}^{x}\mathrm{laguerreL}\left(-n-a-1,a,-x\right)$`

• If all arguments are numerical and at least one argument 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.