Binomial polynomials mimicking Riemann's zeta function

Abstract

ABSTRACT The (generalized) Mellin transforms of Gegenbauer polynomials have polynomial factors , whose zeros all lie on the ‘critical line’ (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2, and S:3/1 binomial coefficient forms. Their ‘critical polynomial’ factors are then identified in terms of hypergeometric functions. Furthermore, we extend these results to a one-parameter family of critical polynomials that possess the functional equation . Normalization yields the rational function whose denominator has singularities on the negative real axis. Moreover, as along the positive real axis, from below. For the Chebyshev polynomials, we obtain the simpler S:2/1 binomial form, and with the nth Catalan number, we deduce that and yield odd integers. The results touch on analytic number theory, special function theory, and combinatorics.