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Laguerre polynomials and L function

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


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.


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)


n, a, x

arithmetical expressions

Return Values

Arithmetical expression.

Overloaded By


See Also

MuPAD Functions

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