Congruence properties of Lehmer-Euler numbers

TL;DR

This paper investigates Lehmer's generalized Euler numbers, establishing their congruence properties, recurrence relations, explicit formulas, and connections to Euler and factorial numbers, along with introducing a new polynomial sequence.

Contribution

It provides new congruence properties, recurrence relations, explicit formulas, and a novel polynomial sequence related to Lehmer's generalized Euler numbers.

Findings

Established congruence properties of Lehmer-Euler numbers

Derived recurrence relations and explicit formulas

Connected Lehmer-Euler numbers to Euler and factorial 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, Lehmer's generalized Euler numbers are studied to give certain congruence properties together with recurrence and explicit formulas of the numbers. We also show a new polynomial sequence and its properties. Some identities including Euler and central factorial numbers are obtained.