On Ledin and Brousseau's summation problems

TL;DR

This paper introduces recursive and polynomial methods for evaluating Fibonacci, Lucas, and general second-order sequence sums involving powers and shifts, extending classical results to broader sequences with explicit formulas.

Contribution

It develops new recursive schemes and polynomial expressions for sums of Fibonacci, Lucas, and Horadam sequences, generalizing previous summation formulas to arbitrary second-order sequences.

Findings

Derived recursive schemes for Fibonacci and Lucas sums

Established polynomial forms for Horadam sequence sums

Extended formulas to sequences with arbitrary parameters

Abstract

We develop a recursive scheme, as well as polynomial forms (polynomials in $n$ of degree $m$), for the evaluation of Ledin and Brousseau's Fibonacci sums of the form $S(m,n,r)=\sum_{k=1}^nk^mF_{k + r}$, $T(m,n,r)=\sum_{k=1}^nk^mL_{k + r}$ for non-negative integers $m$ and $n$ and arbitrary integer $r$; $F_j$ and $L_j$ being the $j^{th}$ Fibonacci and Lucas numbers. We also extend the study to a general second order sequence by establishing a recursive procedure to determine $W(m,n,r;a,b,p,q)=\sum_{k=1}^nk^mw_{k+r}$ where $(w_j(a,b;p,q))$ is the Horadam sequence defined by $w_0 = a,\,w_1 = b;\,w_j = pw_{j - 1} - qw_{j - 2}\, (j \ge 2);$ where $a$, $b$, $p$ and $q$ are arbitrary complex numbers, with $p\ne 0$ and $q\ne 0$. An explicit polynomial form for $W(m,n,r;a,b,1,q)$ and more generally for the sum $\mathcal W(m,n,h,r;a,b,p,q) = \sum_{k = 1}^n {V_h^{- k}k^m w_{hk + r}}$, where $(V_j(p,q))=(w_j(2,p;p,q))$, is established. Finally a polynomial form is established for a Ledin-Brousseau sum involving Horadam numbers with subscripts in arithmetic progression.