On Weighted Sums of Horadam Numbers

Abstract

For nonzero integers p,q with p-q≠1, let (U_n )_(n≥0) be the Lucas sequence of the first kind defined by U_n=pU_(n-1)-qU_(n-2) for n≥2 with initial terms U_0=0 and U_1=1. In this paper, we provide explicit formulas for evaluating the weighted sums ∑_(k=1)^n▒k^m  U_k and ∑_(k=1)^n▒k^m  U_(k+r) using binomial coefficients for nonnegative integers m and arbitrary integers r. Furthermore, we extend these summation formulas to the Horadam sequence (W_n )_(n≥0) defined by the same recursion W_n=pW_(n-1)-qW_(n-2) for n≥2 with arbitrary initial terms W_0 and W_1.