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

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.