Abstract
As is well known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate versions of such functions and polynomials, degenerate polylogarithm functions were introduced and degenerate poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some of their properties were investigated. The aim of this paper is to further study some properties of the degenerate poly-Bernoulli polynomials by using three formulas coming from the recently developed ‘λ-umbral calculus’. In more detail, among other things, we represent the degenerate poly-Bernoulli polynomials by higher-order degenerate Bernoulli polynomials and by higher-order degenerate derangement polynomials.
Highlights
Carlitz is the first one who initiated the study of degenerate versions of some special numbers and polynomials, namely the degenerate Bernoulli and Euler polynomials and numbers
We study the properties of degenerate poly-Bernoulli polynomial arising from degenerate polylogarithmic function and give some identities of those polynomials associated with special polynomials which are derived from the properties of λ-Sheffer sequences
3 Conclusion The umbral calculus had been laid as a rigorous foundation by Rota and is based on linear functionals, differential operators, and Sheffer sequences
Summary
Carlitz is the first one who initiated the study of degenerate versions of some special numbers and polynomials, namely the degenerate Bernoulli and Euler polynomials and numbers (see [2]). The motivation for the paper [10] starts from the question that what if the usual exponential function is replaced with the degenerate exponential functions (see (2)). As it turns out, it corresponds to replacing the linear functional with the family of λlinear functionals (see (12)) and the differential operator with the family of λ-differential operators (see (14)). It corresponds to replacing the linear functional with the family of λlinear functionals (see (12)) and the differential operator with the family of λ-differential operators (see (14)) These replacements lead us to defining λ-Sheffer polynomials and degenerate Sheffer polynomials (see (16))
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