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

Pantelimon Stanica
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Abstract

In this paper we find closed form for the generating function of powers of any non-degenerate second-order recurrence sequence, completing a study begun by Carlitz and Riordan in 1962. Moreover, we generalize a theorem of Horadam on partial sums involving such sequences. Also, we find closed forms for weighted (by binomial coefficients) partial sums of powers of any non-degenerate 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.

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Pantelimon Stanica. 2000-10-15. Generating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences. https://arxiv.org/abs/math/0010149

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