Multiple Sums Involving the Terms of a General Second Order Sequence of Numbers

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 W₀ = a, W₁ = b; W_j = pW_j−1 − qW_j−2 (j≥2); where a, b, p and q are arbitrary complex numbers, with p ≠ 0, q ≠ 0.