Congruence properties of Lehmer-Euler numbers

TL;DR

This work investigates the arithmetic of Lehmer-Euler numbers $W_n$, generalizations of Euler numbers defined by a cubic-root symmetry. It develops a comprehensive framework: (i) foundational properties and multiple explicit representations of $W_{3n}$, (ii) detailed congruence analyses modulo powers of $3$ using Lucas-type methods and new conjectural stabilization patterns, (iii) incomplete variants $W_{n,\le m}$ and $W_{n,\ge m}$ with accompanying recurrences and closed forms, (iv) a higher-order generalization $W_{r,n}^{(\alpha)}$ that encompasses the original case at $r=3$, and (v) a novel polynomial sequence $\Delta(x,k)$ that connects Euler numbers to central factorial numbers via elegant identities. These contributions illuminate the finer arithmetic structure of Lehmer-Euler numbers and broaden connections to central factorial combinatorics, with potential implications for number-theoretic congruences and combinatorial representations.

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.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Proposition 1
• Theorem 3
• proof
• Lemma 1: Lucas's Theorem
• Theorem 4
• proof
• Proposition 2
• Conjecture 1