A note on degenerate poly-Genocchi numbers and polynomials

• Published in: Advances in difference equations 2020-08, Vol.2020 (1), p.1-14, Article 392
• Main Authors: Kim, Hye Kyung, Jang, Lee-Chae
• Format: Article
• Language: English
• Subjects: Analysis; Applications; Arithmetic; Combinatorial analysis; Degenerate poly-Genocchi polynomials and numbers; Difference and Functional Equations; Functional Analysis; Higher-order Bernoulli polynomials and numbers; Mathematical functions; Mathematics; Mathematics and Statistics; Methods; Modified degenerate polyexponential functions; Ordinary Differential Equations; Partial Differential Equations; Polynomials; Topics in Special Functions and q-Special Functions: Theory; Unipoly functions; Unipoly Genocchi polynomials
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-020-02847-y

Recently, some mathematicians have been studying a lot of degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects. Our research is also interested in this field. In this paper, we introduce a new type of the degenerate poly-Genocchi polynomials and numbers, based on Kim and Kim’s ( J. Math. Anal. Appl. 487(2):124017, 2020 ) modified polyexponential function. The paper is divided into two parts. In Sect.  2 , we consider a new type of the degenerate poly-Genocchi polynomials and numbers constructed from the modified polyexponential function. We also show several combinatorial identities related to the degenerate poly-Genocchi polynomials and numbers. Some of them include the degenerate and other special polynomials and numbers such as the Stirling numbers of the first kind, the degenerate Stirling numbers of the second kind, degenerate Euler polynomials, degenerate Bernoulli polynomials and Bernoulli numbers of order α , etc. In Sect.  3 , we also introduce the degenerate unipoly Genocchi polynomials attached to an arithmetic function by using the degenerate polylogarithm function. We give some new explicit expressions and identities related to degenerate unipoly Genocchi polynomials and special numbers and polynomials.