Partial sums and generating functions for powers of second order sequences with indices in arithmetic progression

TL;DR

This paper derives formulas for partial sums of powers of second order sequences with indices in arithmetic progression, including Lucas and Horadam sequences, expanding the understanding of their generating functions.

Contribution

It provides new evaluations of sums involving powers of Lucas and Horadam sequences with indices in arithmetic progression, generalizing previous partial sum results.

Findings

Derived formulas for sums of powers of Lucas and Horadam sequences

Extended previous partial sum evaluations to more general cases

Clarified limitations in earlier assumptions about initial terms

Abstract

The sums $\sum_{j = 0}^k {u_{rj + s}^{2n}z^j }$, $\sum_{j = 0}^k {u_{rj + s}^{2n-1}z^j }$, $\sum_{j = 0}^k {v_{rj + s}^{n}z^j }$ and $\sum_{j = 0}^k {w_{rj + s}^{n}z^j }$ are evaluated; where $n$ is any positive integer, $r$, $s$ and $k$ are any arbitrary integers, $z$ is arbitrary, $(u_i)$ and $(v_i)$ are the Lucas sequences of the first kind, and of the second kind, respectively; and $(w_i)$ is the Horadam sequence. Pantelimon St\uanic\ua set out to evaluate the sum $\sum_{j = 0}^k {w_j^n z^j }$. His solution is not complete because he made the assumption that $w_0=0$, thereby giving effectively only the partial sum for $(u_i)$, the Lucas sequence of the first kind.