Studying degenerate versions of various special polynomials has became an active area of research and has yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of the polylogarithm function, the so-called degenerate polylogarithm function. Then we construct a new type of degenerate Bernoulli polynomial and number, the so-called degenerate poly-Bernoulli polynomial and number, by using the degenerate polylogarithm function, and derive several properties concerning the degenerate poly-Bernoulli numbers.

In this paper, we introduce polyexponential functions as an inverse to the polylogarithm functions, construct type 2 poly-Bernoulli polynomials by using this and derive various properties of type 2...

Let \(Y\) be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of Bernoulli polynomials and Euler polynomials, namely the probabilistic Bernoulli polynomials associated \(Y\) and the probabilistic Euler polynomials associated with \(Y\). Also, we introduce the probabilistic \(r\)-Stirling numbers of the second associated \(Y\), the probabilistic two variable Fubini polynomials associated \(Y\), and the probabilistic poly-Bernoulli polynomials associated with \(Y\). We obtain some properties, explicit expressions, certain identities and recurrence relations for those polynomials. As special cases of \(Y\), we treat the gamma random variable with parameters \(\alpha,\beta > 0\), the Poisson random variable with parameter \(\alpha >0\), and the Bernoulli random variable with probability of success \(p\). DOI 10.1134/S106192084010072

The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate Stirling numbers of both kinds associated with degenerate hyperharmonic numbers and also with degenerate Bernoulli, degenerate Euler, degenerate Bell, and degenerate Fubini polynomials.