Generating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences

摘要

In this paper we find closed form for the generating function of powers ofany non-degenerate second-order recurrence sequence, completing a study begunby Carlitz and Riordan in 1962. Moreover, we generalize a theorem of Horadam onpartial sums involving such sequences. Also, we find closed forms for weighted(by binomial coefficients) partial sums of powers of any non-degeneratesecond-order recurrence sequences. As corollaries we give some known andseemingly unknown identities and derive some very interesting congruencerelations involving Fibonacci and Lucas sequences.