## Laguerre Polynomial

Solutions to the Laguerre Differential Equation with are called Laguerre polynomials. The Laguerre polynomials are illustrated above for and , 2, ..., 5.

The Rodrigues formula for the Laguerre polynomials is

 (1)

and the Generating Function for Laguerre polynomials is

 (2)

A Contour Integral is given by
 (3)

The Laguerre polynomials satisfy the Recurrence Relations
 (4)

(Petkovsek et al. 1996) and
 (5)

The first few Laguerre polynomials are

Solutions to the associated Laguerre Differential Equation with are called associated Laguerre polynomials . In terms of the normal Laguerre polynomials,

 (6)

The Rodrigues formula for the associated Laguerre polynomials is
 (7) (8)

and the Generating Function is

 (9)

The associated Laguerre polynomials are orthogonal over with respect to the Weighting Function .
 (10)

where is the Kronecker Delta. They also satisfy
 (11)

Recurrence Relations include

 (12)

and
 (13)

The Derivative is given by
 (14)

In terms of the Confluent Hypergeometric Function,

 (15)

An interesting identity is
 (16)

where is the Gamma Function and is the Bessel Function of the First Kind (Szegö 1975, p. 102). An integral representation is
 (17)

for , 1, ...and . The Discriminant is
 (18)

(Szegö 1975, p. 143). The Kernel Polynomial is

 (19)

where is a Binomial Coefficient (Szegö 1975, p. 101).

The first few associated Laguerre polynomials are

References

Abramowitz, M. and Stegun, C. 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.

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.

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.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 61-62, 1996.

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

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.