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

2022/08/22 by Kunle Adegoke, Adegoke, Kunle
Mathematics · Physics and Astronomy · #11B37 #11B39 #Advanced Mathematical Theories #Advanced Mathematical Theories and Applications #Benford’s Law and Fraud Detection #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2209.03763

openalex publication_date 2022/08/22 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28

Abstract

We evaluate the nested sum ∑_a_{n - 1} = c^a_n ∑_a_{n - 2} = c^a_{n - 1} ⋯ ∑_{a0 = c}^a₁ x^a₀ 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.

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