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

In this paper we find closed forms of the generating function ∞ ∑ k=0 U r nx , for powers of any non-degenerate second-order recurrence sequence, Un+1 = aUn+bUn−1, a +4b 6= 0, completing a study began by Carlitz [1] and Riordan [4] in 1962. Moreover, we generalize a theorem of Horadam [3] on partial sums involving such sequences. Also, we find closed forms for weighted (by binomial coefficients) partial sums of powers of any nondegenerate second-order recurrence sequences. As corollaries we give some known and seemingly unknown identities and derive some very interesting congruence relations involving Fibonacci and Lucas sequences.