A Short Note on the Pell-Lucas-Eisenstein Series

TL;DR

This paper introduces Pell-Lucas-Eisenstein series, a new class of series based on Pell-Lucas numbers, demonstrating their convergence and functional equations, expanding the understanding of Eisenstein-like series in number theory.

Contribution

It defines Pell-Lucas-Eisenstein series and proves their convergence and functional equations, providing new insights into Eisenstein-like series using Pell-Lucas numbers.

Findings

Pell-Lucas-Eisenstein series are convergent on their domain.

They satisfy specific functional equations.

The proofs are based on calculations involving Pell-Lucas numbers.

Abstract

In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas-Eisenstein Series. Firstly, we show that the Pell-Lucas-Eisenstein series are convergent on their domain. Afterwards we prove that they satisfy some certain functional equations. Proofs follows from some on calculations on Pell-Lucas numbers.