Laguerre Polynomial

The Laguerre polynomials , for nonnegative integer , are polynomial solutions of polynomial degree to the Laguerre differential equation with and . They are illustrated above for and , 2, ..., 5, and implemented in the Wolfram Language as LaguerreL[n, x].

The first few Laguerre polynomials are

			(1)
			(2)
			(3)
			(4)

The coefficients of , ordered from smallest to largest powers, form the triangle 1; , 1; 2, , 1; , 18, , 1; 24, , ... (OEIS A021009). The absolute values of the normalizing factors are the factorials 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, ... (OEIS A000142), starting at .

The Laguerre polynomials are given by the sum

(5)

where is a binomial coefficient.

The Rodrigues representation for the Laguerre polynomials is

(6)

and the generating function for Laguerre polynomials, convergent for , is

			(7)
			(8)

A contour integral that is commonly taken as the definition of the Laguerre polynomial is given by

(9)

where the positively oriented simple contour encloses the origin in the -plane but not (Arfken 1985, pp. 416 and 722).

For , the Laguerre polynomials satisfy the recurrence relations

(10)

(Petkovšek et al. 1996) and

(11)

Polynomial solutions of polynomial degree to the associated Laguerre differential equation with and , for integer , are called associated Laguerre polynomials (Arfken 1985, p. 726) or, in older literature, Sonine polynomials (Sonine 1880, p. 41; Whittaker and Watson 1990, p. 352).