Binomial Sums Involving Second-Order Linearly Recurrent Sequences

Abstract

Consider the sequences ( U-n : n is an element of N-0 ) and ( V-n : n is an element of N) satisfying the second order linear recurrences U-n = pU(n-1) + Un-2 and V-n = pV (n-1) + Vn-2 with the initial conditions U-0 = 0, U-1 = 1, V-0 = 2, and V-1 = p. We explore the problem of evaluating binomial sums involving products consisting of entries in the U- and V - sequences. We apply a hypergeometric approach, inspired by Dilcher's work on hypergeometric identities for Fibonacci numbers, to obtain many new identities for sums involving U and V and products of binomial coefficients, including a non-hypergeometric analogue of Dixon's binomial identity.