Some Positive Kernels and Bilinear Sums for Hahn Polynomials

Abstract

A five-parameter family of kernels $K_N (i,j;\alpha _1 ,\beta _1 ,\beta _2 ,\beta _3 ,\alpha _2 )$ is constructed by using the Hahn polynomials $Q_N (i;\alpha _1 + \beta _k - 1,\alpha _2 + \beta _2 + \beta _3 - \beta _k - 1,N)$, $k = 1,2$, under the assumption that the real parts of the parameters $\alpha _1 $, $\beta _1 $, $\beta _2 $, $\beta _3 $, $\alpha _1 $ are positive. For real values of these parameters this kernel is shown to be positive. Special limiting kernels are obtained by considering various limiting values of the parameters. Some bilinear formulas for the Hahn and Meixner polynomials are also derived.