GENERALIZATION OF AN IDENTITY INVOLVING THE GENERALIZED FIBONACCI NUMBERS AND ITS APPLICATIONS M. FARROKHI D. G. Abstract. We first generalize an identity involving the generalized Fibonacci numbers and then apply it to establish some general identities concerning special sums. We also give a sufficient condition on a generalized Fibonacci sequence {Un} such that Un is divisible by an arbitrary prime r for some 2 <n ≤ r - 2. 1. Preliminaries The generalized Fibonacci and Lucas numbers are defined, respectively, by the Binet’s formulas, as follows U n (p, q)= α n − β n α − β ,V n (p, q)= α n + β n , where α = 1 2 (p +  p 2 − 4q) and β = 1 2 (p −  p 2 − 4q). The numbers U n (p, q) and V n (p, q) can be defined recursively by U n (p, q)= −qU n−2 (p, q)+ pU n−1 (p, q),V n (p, q)= −qV n−2 (p, q)+ pV n−1 (p, q), for all integers n, where U 0 = 0, U 1 = 1, V 0 = 2 and V 1 = p. Throughout the paper, p and q denote the real numbers, U n and V n stand for U n (p, q) and V n (p, q), respectively, and Δ = p 2 − 4q. A sequence {G n } is said to be a (p, q)-sequence if G n satisfies the recursive relation G n = −qG n−2 + pG n−1 , for all integers n. Clearly, the (p, q)-sequences, which are identified at two consec- utive indices should be equal. It is known that the formula U a+b = −qU a−1 U b + U a U b+1 is valid for any generalized Fibonacci sequence {U n (p, q)} and all integers a, b. We intend to present a generalization of this identity and derive some of its applications, which are the general solutions of some solved and unsolved problems concerning the generalized Fibonacci numbers. In Section 2, we prove our claim and give its generalization. In Section 3, we apply our identity to evaluate some summations involving that of Mansour [4] (see also [5]), the sum of powers of the generalized Fibonacci numbers and etc. In final section, we use our identity to get a divisibility property of the generalized Fibonacci numbers. 2000 Mathematics Subject Classification. Primary 11B39, 05A19; Secondary 11B83. Key words and phrases. Generalized Fibonacci numbers, combinatorial identities. 1