A q-Supercongruence Motivated by Higher-Order Generalized Lehmer-Euler Numbers

TL;DR

This paper introduces a new polynomial linked to higher-order generalized Lehmer-Euler numbers and establishes a q-supercongruence, advancing the understanding of these number generalizations.

Contribution

The paper defines a novel polynomial associated with higher-order Lehmer-Euler numbers and proves a related q-supercongruence, extending previous number theory results.

Findings

Established a new q-supercongruence for the polynomial

Connected higher-order Lehmer-Euler numbers with supercongruence properties

Extended classical results on Euler and Bernoulli numbers

Abstract

Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, we define a new polynomial related to the higher-order generalized Lehmer-Euler numbers and determine its a q-supercongruence.