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Laguerre Polynomial

LaguerreL

The Laguerre polynomials are solutions L_n(x) to the Laguerre differential equation with nu=0. They are illustrated above for x in [0,1] and n=1, 2, ..., 5, and implemented in the Wolfram Language as LaguerreL[n, x].

The first few Laguerre polynomials are

L_0(x)=1
(1)
L_1(x)=-x+1
(2)
L_2(x)=1/2(x^2-4x+2)
(3)
L_3(x)=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; -1, 1; 2, -4, 1; -6, 18, -9 1; 24, -96, ... (OEIS A021009). The leading denominators are 1, -1, 2, -6, 24, -120, 720, -5040, 40320, -362880, 3628800, ... (OEIS A000142).

The Laguerre polynomials are given by the sum

L_n(x)=sum_(k=0)^n((-1)^k)/(k!)(n; k)x^k,
(5)

where (n; k) is a binomial coefficient.

The Rodrigues representation for the Laguerre polynomials is

L_n(x)=(e^x)/(n!)(d^n)/(dx^n)(x^ne^(-x))
(6)

and the generating function for Laguerre polynomials is

g(x,z)=(exp(-(xz)/(1-z)))/(1-z)
(7)
=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

L_n(z)=1/(2pii)∮(e^(-zt/(1-t)))/((1-t)t^(n+1))dt,
(9)

where the contour gamma encloses the origin but not the point z=1 (Arfken 1985, pp. 416 and 722).

The Laguerre polynomials satisfy the recurrence relations

(n+1)L_(n+1)(x)=(2n+1-x)L_n(x)-nL_(n-1)(x)
(10)

(Petkovšek et al. 1996) and

xL_n^'(x)=nL_n(x)-nL_(n-1)(x).
(11)

Solutions to the associated Laguerre differential equation with nu!=0 and k an integer are called associated Laguerre polynomials 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, Multivariate Laguerre Polynomial, Orthogonal Polynomials

Related 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/

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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 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'intégrale 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.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 L_n(x)." 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.

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Laguerre Polynomial

Cite this as:

Weisstein, Eric W. "Laguerre Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaguerrePolynomial.html

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