Laguerre Polynomial
The Laguerre polynomials are solutions ๐ L_n(x)
to the Laguerre
differential equation with ๐ nu=0
. They are illustrated above for ๐ x in [0,1]
and ๐ n=1
, 2, ..., 5, and implemented in the Wolfram
Language as [n,
x].
![๐ x in [0,1]](https://mathworld.wolfram.com/images/equations/LaguerrePolynomial/Inline3.svg){kind=link}
The first few Laguerre polynomials are
| ๐ L_0(x) | ๐ = | ๐ 1 |
(1)
|
| ๐ L_1(x) | ๐ = | ๐ -x+1 |
(2)
|
| ๐ L_2(x) | ๐ = | ๐ 1/2(x^2-4x+2) |
(3)
|
| ๐ L_3(x) | ๐ = | ๐ 1/6(-x^3+9x^2-18x+6). |
(4)
|
When ordered from smallest to largest powers and with the denominators factored out, the triangle of nonzero coefficients is 1; ๐ -1
, 1; 2, ๐ -4
, 1; ๐ -6
, 18, ๐ -9
1; 24, ๐ -96
, ... (OEIS A021009).
The leading denominators are 1, ๐ -1
, 2, ๐ -6
, 24, ๐ -120
, 720, ๐ -5040
, 40320, ๐ -362880
, 3628800, ... (OEIS A000142).
The Laguerre polynomials are given by the sum
where ๐ (n; k)
is a binomial coefficient.
The Rodrigues representation for the Laguerre polynomials is
and the generating function for Laguerre polynomials is
| ๐ g(x,z) | ๐ = | ๐ (exp(-(xz)/(1-z)))/(1-z) |
(7)
|
| ๐ Image | ๐ = | ๐ 1+(-x+1)z+(1/2x^2-2x+1)z^2+(-1/6x^3+3/2x^2-3x+1)z^3+.... |
(8)
|
A contour integral that is commonly taken as the definition of the Laguerre polynomial is given by
where the contour ๐ gamma
encloses the origin but not the point ๐ z=1
(Arfken 1985, pp. 416 and 722).
The Laguerre polynomials satisfy the recurrence relations
(Petkovลกek et al. 1996) and
Solutions to the associated Laguerre differential equation with ๐ nu!=0
and ๐ k
an integer are called associated
Laguerre polynomials ๐ L_n^k(x)
(Arfken 1985, p. 726) or, in older literature,
Sonine polynomials (Sonine 1880, p. 41; Whittaker and Watson 1990, p. 352).
See also
Associated Laguerre Polynomial, Laguerre Differential Equation, Orthogonal PolynomialsRelated Wolfram sites
http://functions.wolfram.com/Polynomials/LaguerreL/, http://functions.wolfram.com/Polynomials/LaguerreL3/, http://functions.wolfram.com/HypergeometricFunctions/LaguerreLGeneral/, http://functions.wolfram.com/HypergeometricFunctions/LaguerreL3General/Explore with Wolfram|Alpha
More things to try:
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." ยง6.2 in Special Functions. Cambridge, England: Cambridge University Press, pp. 282-293, 1999.Arfken, G. "Laguerre Functions." ยง13.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 721-731, 1985.Chebyshev, P. L. "Sur le dรฉveloppement des fonctions ร une seule variable." Bull. Ph.-Math., Acad. Imp. Sc. St. Pรฉtersbourg 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." ยง1.11 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its ๐ q
-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 47-49, 1998.Laguerre, E. de. "Sur l'intรฉgrale ๐ int_x^(+infty)x^(-1)e^(-x)dx
." Bull. Soc. math. France 7, 72-81, 1879. Reprinted in Oeuvres, Vol. 1. New York: Chelsea, pp. 428-437, 1971.Petkovลกek, 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." ยง3.1 i The Umbral Calculus. New York: Academic Press, pp. 108-113, 1984.Rota, G.-C.; Kahaner, D.; Odlyzko, A. "Laguerre Polynomials." ยง11 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 dรฉveloppement des fonctions continues en sรฉries." Math. Ann. 16, 1-80, 1880.Spanier, J. and Oldham, K. B. "The Laguerre Polynomials ๐ L_n(x)
." Ch. 23 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 209-216, 1987.Szegรถ, 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.
Referenced on Wolfram|Alpha
Laguerre PolynomialCite this as:
Weisstein, Eric W. "Laguerre Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LaguerrePolynomial.html