On the Integral Representations of the $k$-Pell and $k$-Pell-Lucas Numbers

TL;DR

This paper derives integral representations for the $k$-Pell and $k$-Pell-Lucas numbers, establishing identities and connections with classical sequences using Binet's formulas and elementary calculus.

Contribution

It introduces integral representations for $k$-Pell and $k$-Pell-Lucas numbers, linking them to well-known sequences and providing new identities.

Findings

Integral representations for $k$-Pell and $k$-Pell-Lucas numbers derived.

Established identities connecting these numbers with Fibonacci and Lucas sequences.

Validated representations through elementary integral calculus.

Abstract

In this paper, the integral representations of the $k$-Pell and $k$-Pell-Lucas numbers are presented. Using Binet's formulas for these numbers, we obtain a number of identities and use elementary integral calculus to confirm their integral representations.Our results are also deduced and related to the Fibonacci, Lucas, Pell, and Pell-Lucas numbers.