A study of degenerate Bernoulli and Euler numbers via operators

Technical Summary: A Study of Degenerate Bernoulli and Euler Numbers via Operators

Problem Statement
The paper addresses the need for explicit expressions and relationships concerning degenerate versions of special numbers, specifically the degenerate Bernoulli numbers ($\beta_{n,\lambda}$), degenerate Euler numbers ($E_{n,\lambda}$), and type 2 degenerate Euler numbers ($E^*_{n,\lambda}$). While degenerate versions of these polynomials and numbers have been studied using various tools (generating functions, umbral calculus, $p$-adic analysis), this work focuses on the utility of specific differential operators in deriving these explicit forms and establishing connections between them.

Methodology
The core methodology involves the introduction and application of two specific classes of differential operators acting on functions of a real variable $x$:

1. The operator $\left(x^{1-\lambda} \frac{d}{dx}\right)^n$.
2. The operator $x^{n\lambda} \left(x^{1-\lambda} \frac{d}{dx}\right)^n$, which the authors identify as the degenerate version of the operator $\left(x \frac{d}{dx}\right)^n$, denoted as $\left(x \frac{d}{dx}\right)_{n,\lambda}$.

The authors utilize these operators through the following steps:

• Operator Expansion: They derive explicit expansions of these operators in terms of degenerate Stirling numbers of the second kind, $\left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\}_\lambda$, and standard derivatives.
• Application to Generating Functions: The operators are applied to specific generating functions associated with the degenerate numbers. For instance, the operator $\left(x^{1-\lambda} \frac{d}{dx}\right)^n$ is applied to the function $\frac{1}{1+e_\lambda(z)}$ (where $e_\lambda(z)$ is the degenerate exponential) to extract coefficients related to $E_{n,\lambda}$.
• Variable Substitution: A key technique involves the substitution $u = e_\lambda(z)$, which transforms the derivative with respect to $z$ into the operator form involving $u^{1-\lambda} \frac{d}{du}$.
• Evaluation at Zero: The authors evaluate the resulting expressions at $z=0$ (or $x=0$) to isolate the specific degenerate numbers.
• Recursive and Combinatorial Identities: The paper employs identities involving the degenerate falling factorials $(x)_{n,\lambda}$ and binomial inversion to solve for coefficients in polynomial expansions.

Key Contributions and Results

1. Operator Theory and Identities:

   • The paper establishes the expansion $\left(x^{1-\lambda} \frac{d}{dx}\right)^n = \sum_{l=0}^n \left\{ \begin{smallmatrix} n \\ l \end{smallmatrix} \right\}_\lambda x^{l-n\lambda} \left(\frac{d}{dx}\right)^l$.
   • It derives a recurrence relation for the degenerate Stirling numbers of the second kind: $\left\{ \begin{smallmatrix} n+1 \\ l \end{smallmatrix} \right\}_\lambda = \left\{ \begin{smallmatrix} n \\ l \end{smallmatrix} \right\}_\lambda (l - n\lambda) + \left\{ \begin{smallmatrix} n \\ l-1 \end{smallmatrix} \right\}_\lambda$.
   • A Dobinski-like formula is derived for the degenerate Bell polynomials: $e^x \phi_{n,\lambda}(x) = \sum_{j=0}^\infty \frac{(j)_{n,\lambda}}{j!} x^j$.

2. Explicit Expressions for Degenerate Euler Numbers:
   By applying the operator to $\frac{1}{1+e_\lambda(z)}$ and evaluating at $z=0$, the authors derive an explicit formula for the degenerate Euler numbers:
   $$E_{n,\lambda} = \sum_{k=0}^n \left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\}_\lambda (-1)^k k! 2^{-k}$$

3. Relationship Between Degenerate Bernoulli and Euler Numbers:
   The paper establishes a direct relationship between the degenerate Bernoulli numbers and the degenerate Euler numbers. By analyzing the derivative of $\frac{2z}{e_\lambda^2(z)-1}$ and utilizing the identity $\frac{2z}{e_\lambda^2(z)-1} = \frac{z}{e_\lambda(z)-1} - \frac{z}{e_\lambda(z)+1}$, the authors prove:
   $$2^n \beta_{n, \frac{\lambda}{2}} - \beta_{n,\lambda} = -\frac{n}{2} E_{n-1,\lambda}$$
   This result generalizes a known relationship between classical Bernoulli and Euler numbers (obtained by taking $\lambda \to 0$).

4. Explicit Expressions for Type 2 Degenerate Euler Numbers:
   The authors introduce the function $g_{n,\lambda}(u) = u^{n\lambda} \left(u^{1-\lambda} \frac{d}{du}\right)^n \left(\frac{u}{1+u^2}\right)$. By expanding this function in two ways—once via the degenerate falling factorial series and once via a specific polynomial form involving coefficients $a(n,k|\lambda)$—they derive an explicit expression for the type 2 degenerate Euler numbers ($E^*_{n,\lambda}$, denoted as $E_{n,\lambda}$ in the context of type 2 in the text's notation):
   $$E^*_{n,\lambda} = \sum_{k=0}^n 2^{-k} \sum_{j=0}^k (-1)^j \binom{k}{j} (2j+1)_{n,\lambda}$$

Significance and Claims
The paper claims that these specific operators emerge naturally when attempting to determine explicit expressions for degenerate Euler numbers and finding relationships between degenerate Bernoulli and Euler numbers. The significance lies in providing a unified operator-theoretic framework that yields these explicit formulas and relationships without relying solely on generating function manipulations. The authors note that the operators facilitate the transition between different representations of the degenerate numbers, particularly through the substitution $u = e_\lambda(z)$.

The work is presented as a contribution to the broader study of degenerate special polynomials and numbers, offering new combinatorial identities and explicit formulas that reduce to classical results in the limit as $\lambda \to 0$. The authors conclude by stating their intention to continue investigating probabilistic extensions and $\lambda$-analogues of these special numbers.