Octahedrons with equally many lattice points and generalizations

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 Pn(x) = P _m(y) in rational integers x,y, where {P _k(x)} denote special Meixner polynomials {M _k ^(β, ^c)(x)} with β = 1, c = -1. We join the purely algebraic criterion of Y. Bilu and R. F. Tichy (The Diophantine equation f(x) = g(y), Acta Arith. 95 (2000), no. 3, 261-288) with a famous result of P. Erdös and J. L. Selfridge (The product of consecutive integers is never a power, Illinois J. Math. 19 (1975), 292-301) and prove that (euqation presented) with m, n ≥ 3, β ∈ ℤ/{0,-1,-2,-max(n,m) + 1} and c1,c ₂ ∈ ℚ \ {0, 1} only admits a finite number of integral solutions x, y. Some more results on polynomial families in three-term recurrences are presented.