Abstract
In this paper, we define the poly-Bernoulli polynomials of the second kind by using the polyexponential function and find some interesting identities of those polynomials. In addition, we define unipoly-Bernoulli polynomials of the second kind and study some properties of those polynomials.
Highlights
In the book Ars Conjectandi, Bernoulli introduced the Bernoulli number terms of the sum of powers of consecutive integers
The polyexponential function was first studied by Hardy, and Kim and Kim defined polyexponential function as an = ∑∞ n=1 ðxn/n!Þ
In [21], Kim and Kim modified that function which was again called the polyexponential functions as an inverse to the polylogarithm function
Summary
In the book Ars Conjectandi, Bernoulli introduced the Bernoulli number terms of the sum of powers of consecutive integers (see [1, 2]). In [8], Jang and Kim defined type 2 degenerate Bernoulli polynomials and showed that these polynomials could be represented linear combinations of the Stirling numbers of the first and the second kinds, Bernoulli polynomials, and those numbers. The Bernoulli polynomials of the second kind (or the Cauchy polynomials) are defined by the generating function to be bnðxÞ log t ð1. In [17, 19], the authors defined the generalized Stirling numbers of the first and second kinds and generalized binomial coefficients and showed that degenerated special polynomials are represented by linear combinations of those numbers. We define poly-Bernoulli polynomials of the second kind with the polyexponential function and derive some interesting identities between the Stirling numbers of the first kind or the second kind, Bernoulli numbers, Bernoulli numbers of the second kind, and those polynomials. We define unipoly-Bernoulli polynomials of the second kind and derive some interesting identities of those polynomials
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