A Study on the Sum of the Squares of Generalized p-Oresme Numbers: The Sum Formula \(\sum{^n_k{_=}{_0}}x^kW{^2_m{_k}_+{_j}}\)
Yüksel Soykan *
Department of Mathematics, Art and Science Faculty, Zonguldak Bülent Ecevit University, 67100, Zonguldak, Turkey.
*Author to whom correspondence should be addressed.
Abstract
In this paper, closed forms of the sum formulas \(\sum{^n_k{_=}{_0}}x^kW{^2_m{_k}_+{_j}}\) for generalized p-Oresme numbers are presented. As special cases, we give sum formulas of Modified p-Oresme, p-Oresme-Lucas and p-Oresme numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery.
Keywords: Oresme numbers, Modified p-Oresme numbers, p-Oresme-Lucas numbers, p-Oresme numbers, sum formulas