Degenerate Euler- Seidel Method for degenerate Bernoulli, Euler, and Genocchi polynomials

TL;DR

The paper addresses the problem of extending the Euler-Seidel method to a degenerate setting parameterized by $\lambda$ and applies it to degenerate Bernoulli, Euler, and Genocchi polynomials. It introduces a degenerate Euler-Seidel matrix with the recurrence $a_{k,n}(x|\lambda)= (1-(k-n)\lambda)a_{k-1,n}(x|\lambda)+a_{k-1,n+1}(x|\lambda)$, and derives $\lambda$-generalized binomial identities tied to degenerate falling and rising factorials. It then links sequences via the generating-function relation $\overline{A}_{\lambda}(x,t)=e_{\lambda}^{1-\lambda}(t) A_{\lambda}(x,t)$ and obtains explicit identities for $\beta_{n,\lambda}$, $\mathcal{E}_{n,\lambda}$, and $\mathcal{G}_{n,\lambda}$, such as $\beta_{n,\lambda}(x+1-\lambda)= n(x-\lambda)_{n-1,\lambda}+\beta_{n,\lambda}(x-\lambda)$, with analogous formulas for the other polynomials. The results recover classical identities in the limit $\lambda\to 0$ and provide a versatile framework for studying broader classes of degenerate sequences in combinatorics and number theory.

Abstract

This paper introduces a degenerate version of the Euler-Seidel method by incorporating a parameter lambda into the classical recurrence relation. We define a degenerate Euler-Seidel matrix associated with an initial sequence and establish corresponding lambda-generalized binomial identities and generating function relations. By applying this method to the degenerate Bernoulli, Euler, and Genocchi polynomials, we derive several new combinatorial identities. This work extends the classical Euler-Seidel method to the domain of degenerate special polynomials and numbers, providing a new framework for studying their properties.

Theorems & Definitions (6)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Remark 3.6