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

TL;DR

This paper derives formulas for nested sums involving general second order sequences, including the Horadam sequence, and evaluates sums with complex parameters, broadening understanding of such sums in mathematical analysis.

Contribution

It introduces new methods to evaluate multiple sums involving general second order sequences, extending previous results to more general recurrence relations and complex parameters.

Findings

Derived explicit formulas for nested sums of second order sequences.

Extended evaluation techniques to the Horadam sequence with complex parameters.

Provided general results applicable to a wide class of recurrence-based sums.

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 $W_0 = a,\,W_1 = b;\,W_j = pW_{j - 1} - qW_{j - 2}\, (j \ge 2)$; where $a$, $b$, $p$ and $q$ are arbitrary complex numbers, with $p\ne 0$, $q\ne 0$.