Diophantine equations involving general Meixner and Krawtchouk polynomials

Abstract

While counting lattice points in octahedra of different dimensions n and m, it is an interesting question to ask, how many octahedra exist containing equally many such points. This gives rise to the Diophantine equation p _n (x) = p _m (y) in rational integers x, y, where {p _k (x)} denote special Meixner polynomials {M ^(β,c) _k (x)} with β = 1, c = −1. In this paper we join the algorithmic criterion of Bilu and Tichy [4] with a famous result of Erdös and Selfridge [6] and prove that the Diophantine equation