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

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=0∞λn(k)(x)zn/n!=(1+z)12(x−k)/(1−z)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)=sech πxpk(x)=∫−∞∞…∫−∞∞sech πx1sech πx2 … sech π(x−x1−…−xk−1)dx1dx2…dxk−1,   k=2,3,…