laguerreL

Generalized Laguerre Function and Laguerre Polynomials

Syntax

laguerreL(n,x)

laguerreL(n,a,x)

Description

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.

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

n — Degree of polynomial

number | vector | matrix | multidimensional array | symbolic number | symbolic vector | symbolic matrix | symbolic function | symbolic multidimensional array

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

x — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic vector | symbolic matrix | symbolic function | symbolic multidimensional array

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

a — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic vector | symbolic matrix | symbolic function | symbolic multidimensional array

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

More About

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

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

In particular, the generalized Laguerre polynomials satisfy this normalization.

$$\langle \mathrm{laguerreL}\left(n,a,x\right),\mathrm{laguerreL}\left(m,a,x\right)\rangle =\{\begin{array}{cc}0& \text{if }n\ne m\\ \frac{\Gamma \left(a+n+1\right)}{n!}& \text{if }n=m.\end{array}$$

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.