which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer.
More generally, the name Laguerre polynomials is used for solutions of
Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonin polynomials, after their inventor Nikolay Yakovlevich Sonin).
The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form
reducing to the closed form of a following section.
The sequence of Laguerre polynomials n! Ln is a Sheffer sequence,
The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space. They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator.
Physicists sometimes use a definition for the Laguerre polynomials which is larger by a factor of n! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)
- 1 The first few polynomials
- 2 Recursive definition, closed form, and generating function
- 3 Generalized Laguerre polynomials
- 4 Multiplication theorems
- 5 Relation to Hermite polynomials
- 6 Relation to hypergeometric functions
- 7 Notes
- 8 References
- 9 External links
The first few polynomials
These are the first few Laguerre polynomials:
Recursive definition, closed form, and generating function
One can also define the Laguerre polynomials recursively, defining the first two polynomials as
and then using the following recurrence relation for any k ≥ 1:
The closed form is
The generating function for them likewise follows,
Polynomials of negative index can be expressed using the ones with positive index:
Generalized Laguerre polynomials
For arbitrary real α the polynomial solutions of the differential equation 
are called generalized Laguerre polynomials, or associated Laguerre polynomials.
One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as
and then using the following recurrence relation for any k ≥ 1:
The simple Laguerre polynomials are the special case α = 0 of the generalized Laguerre polynomials:
The Rodrigues formula for them is
The generating function for them is
Explicit examples and properties of the generalized Laguerre polynomials
- Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as
- When n is an integer the function reduces to a polynomial of degree n. It has the alternative expression
- in terms of Kummer's function of the second kind.
- The closed form for these generalized Laguerre polynomials of degree n is
- derived by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula.
- The first few generalized Laguerre polynomials are:
- Ln(α) has n real, strictly positive roots (notice that is a Sturm chain), which are all in the interval 
- and summarizing by
- where is the Bessel function.
As a contour integral
Given the generating function specified above, the polynomials may be expressed in terms of a contour integral
where the contour circles the origin once in a counterclockwise direction.
The addition formula for Laguerre polynomials:
Laguerre's polynomials satisfy the recurrence relations
They can be used to derive the four 3-point-rules
combined they give this additional, useful recurrence relations
Since is a monic polynomial of degree in , there is the partial fraction decomposition
The second equality follows by the following identity, valid for integer i and n and immediate from the expression of in terms of Charlier polynomials:
For the third equality apply the fourth and fifth identities of this section.
Derivatives of generalized Laguerre polynomials
Differentiating the power series representation of a generalized Laguerre polynomial k times leads to
This points to a special case (α = 0) of the formula above: for integer α = k the generalized polynomial may be written
the shift by k sometimes causing confusion with the usual parenthesis notation for a derivative.
Moreover, this following equation holds
which generalizes with Cauchy's formula to
The derivative with respect to the second variable α has the form,
This is evident from the contour integral representation below.
The generalized Laguerre polynomials obey the differential equation
which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,
where for this equation only.
In Sturm–Liouville form the differential equation is
which shows that Lα
n is an eigenvector for the eigenvalue n.
The generalized Laguerre polynomials are orthogonal over [0, ∞) with respect to the measure with weighting function xα e−x:
which follows from
If denoted the Gamma distribution then the orthogonality relation can be written as
in the associated L2[0, ∞)-space.
Turán's inequalities can be derived here, which is
The following integral is needed in the quantum mechanical treatment of the hydrogen atom,
Let a function have the (formal) series expansion
Further examples of expansions
Monomials are represented as
while binomials have the parametrization
This leads directly to
for the exponential function. The incomplete gamma function has the representation
Relation to Hermite polynomials
The generalized Laguerre polynomials are related to the Hermite polynomials:
where the Hn(x) are the Hermite polynomials based on the weighting function exp(−x2), the so-called "physicist's version."
Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.
Relation to hypergeometric functions
where is the Pochhammer symbol (which in this case represents the rising factorial).
- Nikolay Sonin (1880). "Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries". Math. Ann. 16 (1): 1–80. doi:10.1007/BF01459227.
- A&S p. 781
- A&S p.509
- A&S p.510
- A&S p. 775
- G. Szegő, "Orthogonal polynomials", 4th edition, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975, p. 198.
- D. Borwein, J. M. Borwein, R. E. Crandall, "Effective Laguerre asymptotics", SIAM J. Numer. Anal., vol. 46 (2008), no. 6, pp. 3285-3312 doi:10.1137/07068031X
- A&S equation (22.12.6), p. 785
- W. Koepf, "Identities for families of orthogonal polynomials and special functions.", Integral Transforms and Special Functions 5, (1997) pp.69-102. (Theorem 10)
- A&S p. 774
- C. Truesdell, "On the Addition and Multiplication Theorems for the Special Functions", Proceedings of the National Academy of Sciences, Mathematics, (1950) pp.752-757.
- Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 773, ISBN 978-0486612720, MR 0167642.
- Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
- B. Spain, M.G. Smith, Functions of mathematical physics, Van Nostrand Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials.
- Hazewinkel, Michiel, ed. (2001), "Laguerre polynomials", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Eric W. Weisstein, "Laguerre Polynomial", From MathWorld—A Wolfram Web Resource.
- George Arfken and Hans Weber (2000). Mathematical Methods for Physicists. Academic Press. ISBN 0-12-059825-6.
- S. S. Bayin (2006), Mathematical Methods in Science and Engineering, Wiley, Chapter 3.