Abstract
Recently, Kim et al. (Adv. Differ. Equ. 2020:168, 2020) considered the poly-Bernoulli numbers and polynomials resulting from the moderated version of degenerate polyexponential functions. In this paper, we investigate the degenerate type 2 poly-Bernoulli numbers and polynomials which are derived from the moderated version of degenerate polyexponential functions. Our degenerate type 2 degenerate poly-Bernoulli numbers and polynomials are different from those of Kim et al. (Adv. Differ. Equ. 2020:168, 2020) and Kim and Kim (Russ. J. Math. Phys. 26(1):40–49, 2019). Utilizing the properties of moderated degenerate poly-exponential function, we explore some properties of our type 2 degenerate poly-Bernoulli numbers and polynomials. From our investigation, we derive some explicit expressions for type 2 degenerate poly-Bernoulli numbers and polynomials. In addition, we also scrutinize type 2 degenerate unipoly-Bernoulli polynomials related to an arithmetic function and investigate some identities for those polynomials. In particular, we consider certain new explicit expressions and relations of type 2 degenerate unipoly-Bernoulli polynomials and numbers related to special numbers and polynomials. Further, some related beautiful zeros and graphical representations are displayed with the help of Mathematica.
Highlights
IntroductionCarlitz [1, 2], Kim and Kim [14, 19, 21,22,23], Kim et al [26, 28, 29, 31], Jang et al [7, 8], Muhiuddin et al [37,38,39], Khan et al [10,11,12,13], Sharma et al [41,42,43] introduced and studied various degenerate versions of special polynomials and numbers like degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Daehee polynomials, degenerate
For κ ∈ Z, the modified degenerate polyexponential function [30] is specified by Kim and Kim to be
This article aims to present type 2 degenerate poly-Bernoulli numbers and polynomials arising from moderated degenerate polyexponential functions
Summary
Carlitz [1, 2], Kim and Kim [14, 19, 21,22,23], Kim et al [26, 28, 29, 31], Jang et al [7, 8], Muhiuddin et al [37,38,39], Khan et al [10,11,12,13], Sharma et al [41,42,43] introduced and studied various degenerate versions of special polynomials and numbers like degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Daehee polynomials, degenerate. Degenerate Stirling numbers of the first and second kinds.
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