Binomial Sums Involving Second-Order Linearly Recurrent Sequences

Abstract

Consider the sequences (Un : n ∈ N0) and (Vn : n ∈ N) satisfying the secondorder linear recurrences Un = pUn-1+Un-2 and Vn = pVn-1+Vn-2 with the initial conditions U0 = 0, U1 = 1, V0 = 2, and V1 = 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. © 2024 Fibonacci Association. All rights reserved.