New type degenerate Simsek numbers and related aspects

TL;DR

The paper addresses defining and studying a new-type degenerate Simsek number $y_{1,\alpha}^{*}(n,k;\lambda)$. It introduces the generating function $F_{k}(t;\alpha,\lambda)=\frac{(\lambda e^t+1)_{k,\alpha}}{k!}$ and derives explicit formulas, a derivative formula, a recurrence relation, and integral representations for $y_{1,\alpha}^{*}(n,k;\lambda)$; furthermore, $y_{1,\alpha}^{*}(n,k;\lambda)$ reduces to the classical Simsek numbers when $\alpha=0$. Connections to $S_{1}(n,k)$, $S_{2,\alpha}(n,k)$, Bernoulli numbers of order $k$, and degenerate Stirling and Apostol-Euler numbers are established, revealing a rich network of relations. Numerical results and graphical illustrations validate the theory and illustrate how the parameters $\alpha$ and $\lambda$ control deformation and growth across $n$ and $k$. The work extends the combinatorial-analytic framework around degenerate and Simsek-type numbers and suggests potential applications in discrete mathematics and related fields.

Abstract

In this paper, we introduce a new type degenerate Simsek numbers and their generating function, which are different from degenerate Simsek number studied so far. We establish the explicit formula, recurrence relation and other identities for these numbers. We also derive several interesting expressions and relations between these numbers and certain other special numbers in the literature. In addition, several numerical examples and graphical illustrations are provided to support the theoretical results and to illustrate the behavior of the introduced numbers.

Figures (2)

• Figure 1: Numerical comparison of the degenerate Simsek numbers $y_{1,\alpha}^{*}(n,3;\lambda)$. Panels (a) and (c) illustrate the dependence on $\lambda$ for fixed $\alpha$, while panels (b) and (d) show the effect of varying $\alpha$ for fixed $\lambda$.
• Figure 2: Numerical comparison of the degenerate Simsek numbers $y_{1,\alpha}^{*}(3,k;\lambda)$. Panels (a) and (c) illustrate the dependence on $\lambda$ for fixed $\alpha$, while panels (b) and (d) show the effect of varying $\alpha$ for fixed $\lambda$.

Theorems & Definitions (27)

• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• proof
• Proposition 2.4
• Remark 2.5
• Example 2.6
• Theorem 2.7
• proof
• Theorem 2.8