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International Journal of Mathematics and Mathematical Sciences
Volume 3, Issue 4, Pages 761-771
http://dx.doi.org/10.1155/S0161171280000555

Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals

Department of Mathematics, University of Malaya, Kuala Lumpur 22-11, Malaysia

Received 24 August 1978

Copyright © 1980 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using the technique of canonical expansion in probability theory, a bilinear summation formula is derived for the special case of the Meixner-Pollaczek polynomials {λn(k)(x)} which are defined by the generating function n=0λn(k)(x)zn/n!=(1+z)12(xk)/(1z)12(x+k),|z|<1.

These polynomials satisfy the orthogonality condition pk(x)λm(k)(ix)λn(k)(ix)dx=(1)nn!(k)nδm,n,i=1with respect to the weight function p1(x)=sechπxpk(x)=sechπx1sechπx2sechπ(xx1xk1)dx1dx2dxk1,k=2,3,