Remarks on some associated Laguerre integral results

Motivated essentially by their possible need in a fairly large number of physical and chemical contexts, Mavromatis and Alsssar [l] derived several associated Laguerre integral results by eliminating au unnecessary constraint used in an earlier paper on the subject by Mavromatis [2]. The main object of the present sequel to these recent works is to investigate and apply much more general families of integral formulas, involving products of two or more Laguerre polynomials, which have been considered in the mathematical literature rather extensively. @ 2003 Elsevier Ltd. All rights reserved. Keywords-Laguerre polynomials, Orthogonality property, Generalized hypergeometric function, Noncentral Gamma distributions, Appell functions, Chu-Vaudermonde theorem, Degrees of freedom, Coulomb and oscillator systems, Quantum numbers. In the currently popular notations (see, for example, [3, Chapter V]), the Laguerre polynomials L?)(s) (of order Q! and degree n in z) are defined by The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGPOO07353. H.A. Mavromatis and R.S. Alassar would like to acknowledge the support of the King Fahd University of Petroleum and Minerals in this work. 0893-9659/03/g see front matter @ 2003 Elsevier Ltd. All rights reserved. doi: 10.1016/S0893-9659(03)0015+1 1132 H. M. SRIVASTAVA et al. or, equivalently, by where, as usual, PFq denotes a generalized hypergeometric function with p numerator and q denominator parameters. These polynomials satisfy the following orthogonality property: xQ eP L~)(x)L~)(x) dx = (3) (%(a) > -1; m, n E NJO := N U (0))) where 6,,, is the Kronecker delta and N is the set of positive integers. Throughout this paper, we make use of the following general form of the relatively more familiar binomial coefficients: x 0 x .r(x -t 1) P .P(X-I-1+l)I%+l) = x-p ( > (&cl E Q, so that, obviously, we have x 0 = 1 and x 0 = X(X 1). . . (X n + 1) 0 n n. I (n E N). Making use of a slightly different notation L:(x) for the so-called associated Laguerre polyno mials, where C;(X) = r(0 + n + l)Lt)(z), Mavromatis [2] evaluated the following integral involving the product of two Laguerre polynomials: Jrn xp e+ Lk)(x)L$f)(x) dx = m+a 0 ( >( m n+P;p-l ryp+q > ~~Fz(-m,~+1,~-/3+1;a+1,~---n+1;1) (4) (WP.) > -1; m,n E NO), where (and elsewhere in this paper) it is tacitly assumed that the various parameters involved are so constrained that no zeros appear in the denominator on the right-hand side. Subsequently, Lee [4] deduced the integral formula (4) as an easy consequence of a well-known (rather classical) result involving the product of several Laguerre polynomials (see, for example, [5, p. 260, Problem 2(ii)]; see also [6] for a probabilistic derivation of this general result, which is based mainly upon the moments of certain noncentral Gamma distributions). On the other hand, Lee et al. [7] considered a family of such integrals as (for example) in (4) involving products of Laguerre, Hermite, and other classical orthogonal polynomials, pointed out relevant connections of some of these integral formulas with various known integrals, and also investigated the computational and numerical aspects of the results presented by them (see, for details, [7]). In view of the potential for their usefulness in various physical and chemical contexts, Mavromatis and Alassar [l] deduced many further special cases of the integral formula (4) without an obviously unnecessary parametric restriction by which Mavromatis [2] had earlier constrained (4). Here, in the present sequel to each of these recent works, we begin by recalling the following special case of the aforementioned general (classical) result: