Associated Laguerre Polynomials [PDF]

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Associated 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 generalized Laguerre polynomial of degree n is



(derived equivalently by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula.)  o



The first few generalized Laguerre polynomials are:



 o o



The coefficient of the leading term is (−1)n/n!; The constant term, which is the value at 0, is







The explicit formula allows the generalized Laguerre polynomials to be computed using Horner's method, however, the resulting algorithm is not stable. The following method is stable:



function LaguerreL(n, alpha, x) { L1:= 0; LaguerreL:= 1; for i:= 1 to n { L0:= L1; L1:= LaguerreL; LaguerreL:= ((2* i- 1+ alpha- x)* L1- (i- 1+ alpha)* L0)/ i;} return LaguerreL; }







Ln(α) has n real, strictly positive roots (notice that



is a Sturm chain), which are all in



the interval 



The polynomials' asymptotic behaviour for large n, but fixed α and x > 0, is given by



and



[1]



Recurrence relations Laguerre's polynomials satisfy the recurrence relations



in particular



and



or



.



moreover



They can be used to derive the four 3-point-rules



combined they give this additional, useful recurrence relations



A somewhat curious identity, valid for integer i and n, is



it may be used to derive the partial fraction decomposition



Derivatives of generalized Laguerre polynomials Differentiating the power series representation of a generalized Laguerre polynomial k times leads to



moreover, this following equation holds



which generalizes with Cauchy's formula to



The derivate with respect to the second variable α has the surprising form



The generalized associated Laguerre polynomials obey the differential equation



which may be compared with the equation obeyed by the k-th derivative of the ordinary Laguerre polynomial,



where



for this equation only.



This points to a special case (α



= 0) of the formula above: for integer α = k the generalized polynomial



may be written parenthesis notation for a derivative.



, the shift by k sometimes causing confusion with the usual



Orthogonality The associated Laguerre polynomials are orthogonal over [0, ∞) with respect to the measure with weighting function xα e −x:



which follows from



The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula)



recursively



Moreover,



in the associated L2[0, ∞)-space. The following integral is needed in the quantum mechanical treatment of the hydrogen atom,



Series expansions Let a function have the (formal) series expansion



Then



The series converges in the associated Hilbert space



A related series expansion is



, iff



in particular



which follows from



Secondly,



a consequence derived from



for



.



More and other examples Monomials are represented as



binomials have the parametrization



This leads directly to



and, even more generally,



For β a non-negative integer this simplifies to



for γ



= 0 to



or



Jacobi's theta function has the representation



the Bessel function Jα can be expressed (using an arbitrarily chosen parameter t) as



Gamma function has the parametrization



the lower incomplete Gamma function has the representations



and



The upper incomplete gamma function then is



where 2F1 denotes the hypergeometric function.



Multiplication Theorems Erdélyi gives the following two multiplication theorems [2]



 



As contour integral The polynomials may be expressed in terms of a contour integral



where the contour circles the origin once in a counterclockwise direction.



Relation to Hermite polynomials The generalized Laguerre polynomials are related to the Hermite polynomials:



and



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 The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as



where (a)n is the Pochhammer symbol (which in this case represents the rising factorial).



Relation to Bessel functions In terms of modified Bessel functions (Bessel polynomials) these following relations hold:



or further elaborated