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.

8 pages, no figures, no tables