On the Konhauser sets of biorthogonal polynomials suggested by the Laguerre polynomials (original) (raw)
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A note on the Konhauser sets of biorthogonal polynomials suggested by the Laguerre polynomials
Pacific Journal of Mathematics, 1980
Two new classes of bilateral generating functions are given for the Konhauser polynomials Y"{x\ k), which are biorthogonal to the Eonhauser polynomials Z%(x; k) with respect to the weight function x a e~x over the interval (0, oo), a>-1 and fc=l,2,3,. The bilateral generating functions (1) and (2) below would reduce, when fc=l, to similar results for the generalized Laguerre polynomials L^(x). Furthermore, for k-2, these formulas yield the corresponding properties of the Preiser polynomials. It is also shown how the bilateral generating function (2) can be applied to derive a new generating function for the product YV**(x;k)Zt(y;l), where a, β >-1, k, I = 1, 2, 3, , and n = 0,1, 2, .
Pacific Journal of Mathematics, 1985
The polynomial sets {Y"(x; k)} and { Z"(x; &)}, discussed by Joseph D. E. Konhauser, are biorthogonal over the interval (0, oo) with respect to the weight function x a e~x, where a >-1 and A: is a positive integer. The object of the present note is to develop a fairly elementary method of proving a general multilinear generating function which, upon suitable specializations, yields a number of interesting results including, for example, a multivariable hypergeometric generating function for the multiple sum: involving the Konhauser biorthogonal polynomials;
Some functions that generalize the Krall-Laguerre polynomials
Journal of Computational and Applied Mathematics, 1999
Let L(~) be the (semi-infinite) tridiagonal matrix associated with the three-term recursion relation satisfied by the 1 z~e-Z Laguerre polynomials, with weight function r~-77gi5 , ~ > -1, on the interval [0, c~[. We show that, when c~ is a positive integer, by performing at most ~ successive Darboux transformations from L , we obtain orthogonal polynomials on [0, cx~[ with 'weight distribution' 1-----!~z~-ke-Z + k r(~-k+l) ~-~j=l sJ 6(k-j)(z)' with 1 ~<k~<ct. We prove that, as a consequence of the rational character of the Darboux factorization, these polynomials are eigenfunctions of a (finite order) differential operator. Our construction calls for a natural bi-infinite extension of these results with polynomials replaced by functions, of which the semi-infinite case is a limiting situation. (~)
Some generating functions of the Laguerre and modified Laguerre polynomials
Applied Mathematics and Computation, 2000
Recently, by applying the familiar group-theoretic method of Louis Weisner (1899± 1988), many authors proved various single-as well as multiple-series generating functions for certain modi®ed Laguerre polynomials. The main object of the present sequel to these earlier works is to show how readily each of such generating functions can be derived from the corresponding known result for the classical Laguerre polynomials. Many general families of bilinear, bilateral, or mixed multilateral generating functions for the Laguerre (or modi®ed Laguerre) and related polynomials, which are seemingly relevant to the present investigation, are also considered. Ó
A class of bilateral generating functions for certain classical polynomials
Pacific Journal of Mathematics, 1972
In this paper the authors first prove a theorem on bilateral generating 1 relations for a certain sequence of functions. It is then shown how the main result can be applied to derive a large variety of bilateral generating functions for the Bessel, Jacobi, Her mite, Laguerre and ultraspherical polynomials, as well as for their various generalizations. Some recent results given by W. A. Al-Salam [1], S. K. Chatterjea [2], M. K. Das [3], S. Saran [6] and the present authors [9] are thus observed to follow fairly easily as special cases of the theorem proved in this paper.
2010
The generating relations of classical Laguerre polynomials ) ( x L n α contain exponential function. The principal object of this paper is to introduce two parameter Laguerre type polynomials ( ) x L n ;
Tamkang Journal of Mathematics, 2014
Recently, Liu et al. [Bilateral generating functions for the Erkucs-Srivastava polynomials and the generalized Lauricella function, Appl.Math.Comput.218 (2012),pp.7685-7693 investigated some various families of bilateral generating functions involving the Erkucs Srivastava polynomials. The aim of this present paper is to obtain some bilateral generating functions involving the Erkucs-Srivastava polynomials and three general classes of multivariable polynomials introduced earlier by Srivastava in A contour integral involving Fox's H-function, Indian J.Math.14 (1972), pp.1-6, A multilinear generating function for the Konhauser sets of biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J.Math.117 (1985), pp.183-191] and by Kaanouglu and Ozarslan in Two-sided generating functions for certain class of r-variable polynomials, Mathematical and Computer Modelling 54 (2011), pp.625-631. Special cases involving the (Srivastava-Daoust) generalized Lauricella functions ...