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) |
(2) |
(3) |
(4) |
(5) |
Solutions to the associated Laguerre Differential Equation with are called associated Laguerre
polynomials . In terms of the normal Laguerre polynomials,
(6) |
(7) | |||
(8) |
(9) |
(10) |
(11) |
Recurrence Relations include
(12) |
(13) |
(14) |
In terms of the Confluent Hypergeometric Function,
(15) |
(16) |
(17) |
(18) |
(19) |
The first few associated Laguerre polynomials are
See also Sonine Polynomial
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.
© 1996-9 Eric W. Weisstein