[Paper Review] Fully degenerate poly-Bernoulli numbers and polynomials

Dae San Kim, Taekyun Kim|arXiv (Cornell University)|May 26, 2015

Advanced Mathematical Identities10 references3 citations

TL;DR

This paper introduces fully degenerate poly-Bernoulli numbers and polynomials as a new generalization of classical poly-Bernoulli numbers using degenerate exponential functions and polylogarithm functions. It derives explicit identities and integral representations via umbral calculus, showing that the classical poly-Bernoulli polynomials emerge as a limit when the degeneracy parameter λ approaches zero.

ABSTRACT

In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and investigate some properties of these polynomials and numbers. From our properties, we derive some identities for the fully degenerate poly-Bernoulli numbers and polynomials.

Motivation & Objective

• To define a new class of degenerate poly-Bernoulli numbers and polynomials by generalizing the classical poly-Bernoulli polynomials through degenerate exponential functions.
• To investigate the algebraic and analytic properties of these new polynomials using umbral calculus and generating functions.
• To establish connections between the fully degenerate poly-Bernoulli polynomials and classical poly-Bernoulli polynomials via a limiting process as λ → 0.
• To derive explicit identities and integral representations for the fully degenerate poly-Bernoulli numbers and polynomials.
• To express classical poly-Bernoulli polynomials as linear combinations of the fully degenerate ones, providing a new expansion framework.

Proposed method

• Define the generating function for fully degenerate poly-Bernoulli polynomials using the degenerate exponential function and the polylogarithm function: $\frac{\mathrm{Li}_{k}\left(1-\left(1+\lambda t\right)^{-\frac{1}{\lambda}}\right)}{1-\left(1+\lambda t\right)^{-\frac{1}{\lambda}}}\left(1+\lambda t\right)^{\frac{x}{\lambda}} = \sum_{n=0}^{\infty}\beta_{n,\lambda}^{\left(k\right)}\left(x\right)\frac{t^{n}}{n!}$.
• Use umbral calculus techniques to derive operational identities, including the action of differential operators on the polynomials.
• Establish a connection between the polynomials and the classical poly-Bernoulli polynomials via the limit $\lim_{\lambda \to 0} \beta_{n,\lambda}^{\left(k\right)}\left(x\right) = B_{n}^{\left(k\right)}\left(x\right)$.
• Derive an integral representation using the operator $\frac{e^{yt}-1}{t}$, leading to $\left\langle \frac{e^{yt}-1}{t} \middle| \beta_{n,\lambda}^{\left(k\right)}\left(x\right) \right\rangle = \int_{x}^{x+y} \beta_{n,\lambda}^{\left(k\right)}\left(u\right) du$.
• Apply the inversion formula using the dual basis to express arbitrary polynomials in $\mathbb{P}_n$ as linear combinations of $\beta_{m,\lambda}^{\left(k\right)}\left(x\right)$, with coefficients derived from the inner product with $\frac{1-e^{-t}}{\mathrm{Li}_k(1-e^{-t})} \left( \frac{1}{\lambda}(e^{\lambda t}-1) \right)^m$.
• Use Stirling numbers of the second kind $S_2(n,m)$ to express the expansion coefficients, yielding $a_m = \lambda^{n-m} S_2(n,m)$ in the expansion of $B_n^{(k)}(x)$.

Experimental results

Research questions

• RQ1 How can the classical poly-Bernoulli polynomials be generalized through a degeneracy parameter λ in the exponential function?
• RQ2 What are the structural and operational identities satisfied by the newly defined fully degenerate poly-Bernoulli polynomials?
• RQ3 How do the fully degenerate poly-Bernoulli polynomials relate to the classical poly-Bernoulli polynomials in the limit as λ → 0?
• RQ4 Can the classical poly-Bernoulli polynomials be expressed as linear combinations of the fully degenerate ones, and if so, what are the coefficients?
• RQ5 What integral and differential identities can be derived for the fully degenerate poly-Bernoulli polynomials using umbral calculus?

Key findings

• The fully degenerate poly-Bernoulli polynomials $\beta_{n,\lambda}^{\left(k\right)}\left(x\right)$ are defined via a generating function involving the degenerate exponential and polylogarithm functions.
• The limit $\lim_{\lambda \to 0} \beta_{n,\lambda}^{\left(k\right)}\left(x\right) = B_{n}^{\left(k\right)}\left(x\right)$ confirms that the classical poly-Bernoulli polynomials are recovered in the non-degenerate limit.
• An integral representation is derived: $\left\langle \frac{e^{t}-1}{t} \middle| \beta_{n,\lambda}^{\left(k\right)}\left(x\right) \right\rangle = \int_{0}^{1} \beta_{n,\lambda}^{\left(k\right)}\left(u\right) du = \sum_{l=0}^{n} \sum_{m=0}^{l} \binom{l}{m} \binom{n}{l} \lambda^{l-m} b_{l-m} \beta_{n-l,\lambda}^{\left(k\right)} \frac{(1|\lambda)_{m+1}}{m+1}$.
• The classical poly-Bernoulli polynomial $B_n^{(k)}(x)$ is expressed as a linear combination of the fully degenerate ones: $B_n^{(k)}(x) = \sum_{m=0}^{n} \lambda^{n-m} S_2(n,m) \beta_{m,\lambda}^{\left(k\right)}\left(x\right)$, where $S_2(n,m)$ are Stirling numbers of the second kind.
• The coefficients in the expansion of $B_n^{(k)}(x)$ in terms of $\beta_{m,\lambda}^{\left(k\right)}\left(x\right)$ are explicitly given by $a_m = \lambda^{n-m} S_2(n,m)$, derived using umbral calculus and inner product identities.