Number Theory


Sums of Powers of the Terms of Lucas Sequences with Indices in Arithmetic Progression

Authors: Kunle Adegoke

We evaluate the sums $\sum_{j=0}^k{u_{rj+s}^{2n}\,z^j}$, $\sum_{j=0}^k{u_{rj+s}^{2n-1}\,z^j}$ and $\sum_{j=0}^k{v_{rj+s}^{n}\,z^j}$, where $r$, $s$ and $k$ are any integers, $n$ is any nonnegative integer, $z$ is arbitrary and $(u_n)$ and $(v_n)$ are the Lucas sequences of the first kind and of the second kind, respectively. As natural consequences we obtain explicit forms of the generating functions for the powers of the terms of Lucas sequences with indices in arithmetic progression. This paper therefore extends the results of P.~Sta\u nic\u a who evaluated $\sum_{j=0}^k{u_{j}^{2n}\,z^j}$ and $\sum_{j=0}^k{u_{j}^{2n-1}\,z^j}$; and those of B. S. Popov who obtained generating functions for the powers of these sequences.

Comments: 5 Pages.

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Submission history

[v1] 2019-04-26 01:20:05

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