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New type degenerate Simsek numbers and related aspects

This paper introduces a new type of degenerate Simsek numbers, establishing their generating functions, explicit formulas, recurrence relations, and connections to other special numbers through theoretical derivations and numerical illustrations.

Original authors: Lahcen Oussi

Published 2026-02-11

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Lahcen Oussi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master chef who has spent years perfecting a classic recipe for a chocolate cake (these are the "Simsek numbers"). Everyone knows this recipe, and it’s used in many different types of desserts.

Now, imagine you decide to experiment. You want to see what happens if you replace the sugar with a special, "deforming" ingredient—like a magical spice that changes the texture of the cake depending on how much you add (this is the "degeneracy parameter $\alpha$ ").

This paper is essentially a scientific "recipe book" for this brand-new, experimental version of the cake.

Here is a breakdown of what the author, Lahcen Oussi, actually did:

1. The "New Recipe" (The New Numbers)

The author didn't just tweak the old recipe; he created a completely new way to mix the ingredients. In math terms, he introduced "New Type Degenerate Simsek Numbers."

If the original Simsek numbers were a standard cake, these new numbers are like a "deformed" cake that can stretch, shrink, or change its density based on a mathematical dial (the $\alpha$ parameter). He provides the "formula" (the generating function) so that other mathematicians can bake this exact same "cake" themselves.

2. The "Kitchen Tools" (Identities and Relations)

When you invent a new recipe, you need to know how it interacts with the tools you already have in your kitchen.

The Connection: The author shows how his new numbers relate to other famous "ingredients" in the math world, like Stirling numbers (which are like the basic flour and eggs of math) and Bernoulli numbers.
The Rules: He establishes "recurrence relations." Think of these as rules like: "If you want to make a double-sized cake, here is exactly how much more spice you need to add." It allows mathematicians to calculate complex numbers by building them up from simpler ones.

3. The "Taste Test" (Numerical and Graphical Results)

A recipe isn't useful if you don't know how it actually tastes. The author performs a "taste test" by:

Creating Tables: He calculates specific values to show exactly what the numbers look like when you plug in different amounts of "spice" ( $\alpha$ ) or "sweetness" ( $\lambda$ ).
Drawing Maps (Graphs): He creates visual charts. Imagine a mountain range where the height of the mountain represents the value of the number. He shows that as you increase certain ingredients, the "mountains" grow much taller and steeper. This helps people "see" the math instead of just staring at endless strings of digits.

Why does this matter? (The "So What?")

You might ask, "Who cares about a new kind of math cake?"

In the real world, these "special numbers" are the hidden gears inside the machines of science. They are used to model things in quantum mechanics (the tiny world of atoms), biology (how populations grow), and probability (predicting the future). By creating this new "recipe," the author has given scientists a new tool to model complex, "deforming" systems that the old recipes couldn't quite handle.

In short: The author has invented a new mathematical "flavor" and provided the complete manual on how to use it, how it relates to everything else, and what it looks like in action.

Technical Summary: New Type Degenerate Simsek Numbers and Related Aspects

Author: Lahcen Oussi
Subject Area: Combinatorics, Special Functions, Discrete Mathematics

1. Problem Statement

The paper addresses the expansion of special number families within the framework of degenerate mathematics. While Simsek numbers (introduced by Simsek in 2018) and their previously known degenerate versions exist in the literature, the author identifies a gap: the lack of a specific type of degenerate Simsek numbers defined through a different generating function structure. The objective is to introduce this new class of numbers, establish their fundamental properties, and map their relationships to other well-known combinatorial sequences.

2. Methodology

The author employs generating function analysis as the primary methodological tool. The research follows a structured mathematical derivation process:

Definition: The new numbers, denoted as $y^*_{1,\alpha}(n, k; \lambda)$ , are defined via a new generating function $F_k(t; \alpha, \lambda) = \frac{(\lambda e^t + 1)^{k,\alpha}}{k!}$ , utilizing the degenerate exponential function $e_{\alpha}(t) = (1+\alpha t)^{1/\alpha}$ .
Algebraic Manipulation: The author uses Taylor expansions, the properties of falling factorials, and the inversion formulas of Stirling numbers to derive explicit identities.
Comparative Analysis: By manipulating the newly defined generating functions, the author relates them to existing sequences (Bernoulli, Stirling, and Apostol-Euler numbers) through coefficient comparison and differential/integral operators.
Computational Validation: The theoretical results are verified using numerical tables and graphical modeling to observe the behavior of the numbers across varying parameters ( $n, k, \alpha, \lambda$ ).

3. Key Contributions

The paper introduces several novel mathematical constructs and identities:

New Class of Numbers: The definition of $y^*_{1,\alpha}(n, k; \lambda)$ , which reduces to the classical Simsek numbers as the degeneracy parameter $\alpha \to 0$ .
Explicit Formulas: Derivation of multiple explicit representations, including those involving Stirling numbers of the first kind ( $S_1$ ) and Bernoulli numbers of order $k$ ( $B_n^{(k)}$ ).
Recurrence and Differential Relations: The establishment of recurrence relations with respect to both the index $k$ and the order $n$ , as well as derivative and integral formulas for the associated generating functions $\phi^*_n(x|\alpha, \lambda)$ .
Inter-sequence Connectivity: A comprehensive mapping that links these new numbers to:
- Stirling numbers of the first and second kind.
- Degenerate Stirling numbers of the second kind.
- Apostol-Euler numbers of order $k$ .
- Classical Simsek numbers.

4. Key Results

Theorem 2.3 & 2.4: Provides the explicit closed-form expressions, allowing for direct computation without recursive steps.
Theorem 2.7 & 2.8: Establishes the bridge between the new Simsek numbers and the degenerate Stirling numbers, proving they are not isolated entities but part of a larger combinatorial ecosystem.
Numerical/Graphical Findings:
- The numbers exhibit polynomial dependence on the parameters $\alpha$ and $\lambda$ .
- As the order $n$ increases, the numbers grow rapidly, with the growth rate being heavily influenced by $\lambda$ .
- The parameter $\alpha$ acts as a "deformation" parameter, altering the structural behavior of the sequence as $k$ and $n$ increase.

5. Significance

This research contributes to the field of enumerative combinatorics and special functions. By introducing a new degenerate variant, the paper provides fresh tools for solving complex problems in mathematical physics, quantum mechanics, and probability theory where special numbers are frequently employed. The ability to express these new numbers through existing sequences (like Stirling and Bernoulli) makes them highly useful for researchers looking to apply degenerate models to discrete mathematical problems.