Abstract
In this paper, we consider the degenerate poly-Bernoulli polynomials. We present several explicit formulas and recurrence relations for these polynomials. Also, we establish a connection between our polynomials and several known families of polynomials.
Highlights
The degenerate Bernoulli polynomials βn(λ, x) (λ = ) were introduced by Carlitz [ ] and rediscovered by Ustinov [ ] under the name Korobov polynomials of the second kind
We observe that limλ→ βn(λ, x) = Bn(x), where Bn(x) is the nth ordinary Bernoulli polynomial
Where Lik(x) (k ∈ Z) is the classical polylogarithm function given by Lik(x) =
Summary
The degenerate Bernoulli polynomials βn(λ, x) (λ = ) were introduced by Carlitz [ ] and rediscovered by Ustinov [ ] under the name Korobov polynomials of the second kind They are given by the generating function For = λ ∈ C and k ∈ Z, the degenerate poly-Bernoulli polynomials Pβn(k)(λ, x) are defined by Kim and Kim to be. The goal of this paper is to use umbral calculus to obtain several new and interesting identities of degenerate poly-Bernoulli polynomials. We denote the algebra of polynomials in a single variable x over C by and the vector space of all linear functionals on by ∗. Λ eλt – In this paper, we will use umbral calculus in order to derive some properties, explicit formulas, recurrence relations, and identities as regards the degenerate poly-Bernoulli polynomials. We establish a connection between our polynomials and several known families of polynomials
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