Solutions to the associated Laguerre differential equation with  and  an 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). Associated Laguerre polynomials are implemented in the Wolfram Language as LaguerreL[n, k, x]. In terms of the unassociated Laguerre polynomials,

(1)

The Rodrigues representation for the associated Laguerre polynomials is

			(2)
			(3)
			(4)
			(5)

where  is a Whittaker function.

The associated Laguerre polynomials are a Sheffer sequence with

			(6)
			(7)

giving the generating function

			(8)
			(9)

where the usual factor of  in the denominator has been suppressed (Roman 1984, p. 31). Many interesting properties of the associated Laguerre polynomials follow from the fact that  (Roman 1984, p. 31).

The associated Laguerre polynomials are given explicitly by the formula

(10)

where  is a binomial coefficient, and have Sheffer identity

(11)

(Roman 1984, p. 31).

The associated Laguerre polynomials are orthogonal over  with respect to the weighting function ,

(12)

where  is the Kronecker delta. They also satisfy

(13)

Recurrence relations include

(14)

and

(15)

The derivative is given by

			(16)
			(17)

An interesting identity is

(18)

where  is the gamma function and  is the Bessel function of the first kind (Szegö 1975, p. 102). An integral representation is

(19)

for , 1, ...and . The polynomial discriminant is

(20)

(Szegö 1975, p. 143). The kernel polynomial is

(21)

where  is a binomial coefficient (Szegö 1975, p. 101).

The first few associated Laguerre polynomials are

			(22)
			(23)
			(24)
			(25)

A generalization of the associated Laguerre polynomial to  not necessarily an integer is called a Laguerre function (Arfken 1985, p. 726) or a generalized Laguerre function (Abramowitz and Stegun 1972, p. 775). These generalized Laguerre polynomial can be defined as

(26)

where  is the Pochhammer symbol and  is a confluent hypergeometric function of the first kind (Koekoek and Swarttouw 1998). They are implemented in the Wolfram Language as LaguerreL[n, alpha, x].

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.

Andrews, G. E.; Askey, R.; and Roy, R. "Laguerre Polynomials." §6.2 in Special Functions. Cambridge, England: Cambridge University Press, pp. 282-293, 1999.

Arfken, G. "Laguerre Functions." §13.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 721-731, 1985.

Chebyshev, P. L. "Sur le développement des fonctions à une seule variable." Bull. Ph.-Math., Acad. Imp. Sc. St. Pétersbourg 1, 193-200, 1859.

Chebyshev, P. L. Oeuvres, Vol. 1. New York: Chelsea, pp. 499-508, 1987.

Iyanaga, S. and Kawada, Y. (Eds.). "Laguerre Functions." Appendix A, Table 20.VI in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1481, 1980.

Koekoek, R. and Swarttouw, R. F. "Laguerre." §1.11 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its -Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 47-49, 1998.

Laguerre, E. de. "Sur l'intégrale ." Bull. Soc. math. France 7, 72-81, 1879. Reprinted in Oeuvres, Vol. 1. New York: Chelsea, pp. 428-437, 1971.

Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 61-62, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

Roman, S. "The Laguerre Polynomials." §3.1 i The Umbral Calculus. New York: Academic Press, pp. 108-113, 1984.

Rota, G.-C.; Kahaner, D.; Odlyzko, A. "Laguerre Polynomials." §11 in "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684-760, 1973.

Sansone, G. "Expansions in Laguerre and Hermite Series." Ch. 4 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 295-385, 1991.

Sloane, N. J. A. Sequences A000142/M1675 and A021009 in "The On-Line Encyclopedia of Integer Sequences."

Sonine, N. J. "Sur les fonctions cylindriques et le développement des fonctions continues en séries." Math. Ann. 16, 1-80, 1880.

Spanier, J. and Oldham, K. B. "The Laguerre Polynomials ." Ch. 23 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 209-216, 1987.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.

Whittaker, E. T. and Watson, G. N. Ch. 16, Ex. 8 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 352, 1990.