Multiple sums involving the terms of a general second order sequence of numbers

Abstract

We evaluate the nested sum ${\sum }_{a_{n-1}=c}^{a_n}{\sum }_{a_{n-2}=c}^{a_{n-1}}\cdots {\sum }_{a_0=c}^{a_1}{x}^{a_0}$ where $a_n$ and $c$ are any integers and $x$ is a real or complex variable. Consequently, we evaluate multiple sums involving the terms of a general second order sequence, the Horadam sequence $(W_j(a,b;p,q))$, defined for all non-negative integers $j$ by the recurrence relation $W_0=a,W_1=b;W_j=p{W}_{j-1}-q{W}_{j-2}(j\ge 2)$; where $a$, $b$, $p$ and $q$ are arbitrary complex numbers, with $p\ne 0$, $q\ne 0$.

arXiv comment16 pages, no figures or tables