Analytical properties of type 2 degenerate poly-Bernoulli polynomials associated with their applications

• Main Authors: Waseem A. Khan, Ghulam Muhiuddin, Abdulghani Muhyi, Deena Al-Kadi
• Format: Article
• Language: English
• Published: SpringerOpen 2021-09-01
• Series: Advances in Difference Equations
• Subjects: Modified degenerate polyexponential functions; Type 2 degenerate Bernoulli polynomials; Type 2 degenerate central Bell polynomials; Unipoly functions
• Online Access: https://doi.org/10.1186/s13662-021-03575-7

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.