Weighted sums of Lucas numbers of the first and second kind

TL;DR

This paper generalizes weighted sums of Fibonacci numbers to Lucas sequences, providing closed-form formulas and new identities for various well-known sequences using Abel's summation method.

Contribution

It extends Wall's 1964 result to Lucas sequences of both kinds and derives new closed-form formulas for weighted sums of multiple classical sequences.

Findings

Closed-form formulas for sums of Lucas sequences of the first and second kind.

Recovery of known identities for Fibonacci and Lucas numbers.

Introduction of new identities for sequences like Pell, Jacobstahl, and Mersenne numbers.

Abstract

In the \textit{Fibonacci Quarterly} in 1964, C.~R.~Wall gave the following weighted sum of generalized Fibonacci numbers: $\sum_{i=1}^n i G_i = n G_{n+2} - G_{n+3} + G_3$, where $\left(G_n\right)_{n \geq 0}$ is defined by the recurrence $G_n = G_{n-1} + G_{n-2}$ with fixed $G_0, G_1 \in \mathbb{Z}$. In this paper, we generalize Wall's result to the Lucas sequences of the first and second kind, $\left(U_n(p,q)\right)_{n \geq 0}$ and $\left(V_n(p,q)\right)_{n \geq 0}$, and give closed forms for $\sum_{i=1}^n i U_i$ and $\sum_{i=1}^n i V_i$ by using Abel's summation by parts method. Moreover, we provide concrete applications, not only recovering the known weighted sums $\sum_{i=1}^n i F_i$ and $\sum_{i=1}^n i L_i$ of Fibonacci and Lucas numbers, respectively, but also add new identities to the literature for eight well-known sequences. In particular, we give closed forms for weighted sums of Lucas sequences of the first kind such as the Pell, balancing, Jacobstahl, and Mersenne numbers, and also Lucas sequences of the second kind such as the companion Pell, double Lucas-balancing, Jacobstahl-Lucas, and Mersenne-Lucas numbers.