Weighted sums of Lucas numbers of the first and second kind
aBa Mbirika
TL;DR
The paper extends C. R. Wall's weighted-sum identity from generalized Fibonacci sequences to the Lucas sequences $U_n(p,q)$ and $V_n(p,q)$ by employing Abel's summation by parts. It first derives closed forms for the sums of consecutive Lucas terms, distinguishing the cases $p-q=1$ and $p-q\neq 1$, and then obtains closed forms for the weighted sums $\sum_{i=1}^n i U_i$ and $\sum_{i=1}^n i V_i$ in those same regimes, introducing the auxiliary functions $\Omega(n)$ and $\Psi(n)$. The results are then specialized to ten well-known Lucas-type sequences, recovering the classical Fibonacci/Lucas formulas and providing new weighted-sum identities for eight additional sequences (Pell, balancing, Jacobstahl, Mersenne families, and their companions). The work culminates in open questions on reverse weighted sums and further generalizations (including higher powers and multi-indexed terms), guiding future research in this area of number theory.
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.