A Multilinear Generating Function for the Konhauser Sets of Biorthogonal Polynomials Suggested by the Laguerre Polynomials

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: 1. Introduction. Joseph D. E. Konhauser ([5]; see also [4]) introduced two interesting classes of polynomials: Y*(x\ k) a polynomial in JC, and Z"(JC; k) a polynomial in JC*, α >-1 and k = 1,2,3, For fc = 1, these polynomials reduce to the classical Laguerre polynomials L^ α) (x), and for k = 2 they were encountered earlier by Spencer and Fano [8] in the study of the penetration of gamma rays through matter and were discussed subsequently by Preiser [7]. Also [5, p. 303]