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Associated Laguerre Polynomial

Associated Laguerre polynomials L_n^k(x), also called generalized Laguerre polynomials, are polynomial solutions to the associated Laguerre differential equation with lambda=n and nu=k, where n is a nonnegative integer. The parameter k need not be an integer. In older literature, they are called 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,

L_n(x)=L_n^0(x).
(1)

The Rodrigues representation for the associated Laguerre polynomials is

L_n^k(x)=(e^xx^(-k))/(n!)(d^n)/(dx^n)(e^(-x)x^(n+k))
(2)
=(-1)^k(d^k)/(dx^k)[L_(n+k)(x)]
(3)
=((-1)^nx^(-(k+1)/2))/(n!)e^(x/2)W_(k/2+n+1/2,k/2)(x)
(4)
=sum_(m=0)^(n)(-1)^m((n+k)!)/((n-m)!(k+m)!m!)x^m,
(5)

where W_(k,m)(x) is a Whittaker function. The form involving d^k/dx^k requires k to be a nonnegative integer.

The associated Laguerre polynomials are a Sheffer sequence with

g(t)=(1-t)^(-k-1)
(6)
f(t)=t/(t-1),
(7)

giving the generating function

g(x,z)=(exp(-(xz)/(1-z)))/((1-z)^(k+1))
(8)
=1+(k+1-x)z1/2[x^2-2(k+2)x+(k+1)(k+2)]z^2+...,
(9)

where the usual factor of n! in the denominator has been suppressed (Roman 1984, p. 31). Many interesting properties of the associated Laguerre polynomials follow from the fact that f^(-1)(t)=f(t) (Roman 1984, p. 31).

The associated Laguerre polynomials are given explicitly by the formula

L_n^k(x)=sum_(i=0)^n1/(i!)(k+n; n-i)(-x)^i,
(10)

where (n; k) is a binomial coefficient, and have Sheffer identity

n!L_n^k(x+y)=sum_(i=0)^n(n; i)i!L_i^k(x)(n-i)!L_(n-i)^(-1)(y)
(11)

(Roman 1984, p. 31).

The associated Laguerre polynomials are orthogonal over [0,infty) with respect to the weighting function x^ke^(-x),

int_0^inftye^(-x)x^kL_n^k(x)L_m^k(x)dx=((n+k)!)/(n!)delta_(mn),
(12)

where delta_(mn) is the Kronecker delta. They also satisfy

int_0^inftye^(-x)x^(k+1)[L_n^k(x)]^2dx=((n+k)!)/(n!)(2n+k+1).
(13)

Recurrence relations include

sum_(nu=0)^nL_nu^k(x)=L_n^(k+1)(x)
(14)

and

L_n^k(x)=L_n^(k+1)(x)-L_(n-1)^(k+1)(x).
(15)

The derivative is given by

d/(dx)L_n^k(x)=-L_(n-1)^((k+1))(x)
(16)
=x^(-1)[nL_n^k(x)-(n+k)L_(n-1)^k(x)].
(17)

An interesting identity is

sum_(n=0)^infty(L_n^k(x))/(Gamma(n+k+1))w^n=e^w(xw)^(-k/2)J_k(2sqrt(xw)),
(18)

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

e^(-x)x^(k/2)L_n^k(x)=1/(n!)int_0^inftye^(-t)t^(n+k/2)J_k(2sqrt(tx))dt
(19)

for n=0, 1, ...and k>-1. The polynomial discriminant is

D_n^k=product_(nu=1)^nnu^(nu-2n+2)(nu+k)^(nu-1)
(20)

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

K_n^k(x,y)=(n+1)/(Gamma(k+1))(n+k; n)^(-1)(L_n^k(x)L_(n+1)^k(y)-L_(n+1)^k(x)L_n(k)(y))/(x-y),
(21)

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

The first few associated Laguerre polynomials are

L_0^k(x)=1
(22)
L_1^k(x)=-x+k+1
(23)
L_2^k(x)=1/2[x^2-2(k+2)x+(k+1)(k+2)]
(24)
L_3^k(x)=1/6[-x^3+3(k+3)x^2-3(k+2)(k+3)x+(k+1)(k+2)(k+3)].
(25)

A hypergeometric function representation is

L_n^alpha(x)=((alpha+1)_n)/(n!)_1F_1(-n;alpha+1;x),
(26)

where (a)_n is the Pochhammer symbol and _1F_1(a;b;x) 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].

For the parameters k=-1/2 and k=1/2, associated Laguerre polynomials give the Hermite polynomials

L_n^(-1/2)(x^2)=((-1)^n)/(2^(2n)n!)H_(2n)(x)
(27)
xL_n^(1/2)(x^2)=((-1)^n)/(2^(2n+1)n!)H_(2n+1)(x).
(28)

This makes the even and odd Hermite polynomials correspond to different weighting functions on [0,infty) (Trott 2006).

For fixed nonnegative integer n and k>-1, associated Laguerre polynomials are also obtained as a limit of Jacobi polynomials,

lim_(beta->infty)P_n^((k,beta))(1-(2x)/beta)=L_n^k(x).
(29)

Das et al. (2026) proved that L_n^k(x) is irreducible over the rational numbers for integers 0<=k<=1000 and n>=1, with exactly the following exceptions. For n=2, the exceptions are k=m^2-2 for integers 2<=m<=31. For n=4, they are k=5 and k=23.

See also

Confluent Hypergeometric Function of the First Kind, Hermite Polynomial, Jacobi Polynomial, Laguerre Polynomial, Whittaker Function

Related Wolfram sites

https://functions.wolfram.com/Polynomials/LaguerreL3/

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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.Das, D. T.; Dutta, S.; Laishram, S.; and Nair, S. G. "Irreducibility of Generalised Laguerre Polynomials for Large Integer Parameters." 7 Oct 2026. https://arxiv.org/abs/2610.10614.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.National Institute of Standards and Technology. "Interrelations and Limit Relations." §18.7 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/18.7.Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 61-62, 1996. https://www2.math.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.Trott, M. "Relationships Among the Orthogonal Polynomials." §2.9 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.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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Associated Laguerre Polynomial

Cite this as:

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

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