On Ledin and Brousseau's summation problems

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.