On the type 2 poly-Bernoulli polynomials associated with umbral calculus

Abstract: Type 2 poly-Bernoulli polynomials were introduced recently with the help of modified polyexponential functions. In this paper, we investigate several properties and identities associated with those polynomials arising from umbral calculus techniques. In particular, we express the type 2 poly-Bernoulli polynomials in terms of several special polynomials, like higher-order Cauchy polynomials, higher-order Euler polynomials, and higher-order Frobenius-Euler polynomials.


Introduction
The poly-Bernoulli polynomials, which are defined with the help of polylogarithm functions, were studied by Kaneko in [1], while the type 2 poly-Bernoulli polynomials, which are defined with the help of modified polyexponential functions, were investigated very recently in [2]. We note that the modified polyexponential functions are inverse to the polylogarithm functions. Thus, it is very natural to replace the polylogarithms by the modified polyexponential functions in the definition of generating function of poly-Bernoulli polynomials. Indeed, the generating function of type 2 poly-Bernoulli polynomials is obtained in this way (see (1), (3)), and hence we may say that it arises in a natural manner.

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The aim or motivation of this paper is to further derive some properties, recurrence relations, and identities related to the type 2 poly-Bernoulli polynomials by using umbral calculus techniques. Especially, those polynomials are represented in terms of some well-known special polynomials. In general, special polynomials and numbers can be studied by employing various different methods including combinatorial methods, generating functions, differential equations, umbral calculus techniques, p-adic analysis, and probability theory.
The outline of this paper is as follows. In Section 1, we give some necessary definitions and some basic facts about umbral calculus. As to definitions, we recall the definitions of polyexponential functions, type 2 poly-Bernoulli polynomials, higher-order Bernoulli polynomials, higher-order Cauchy polynomials, higher-order Euler polynomials, and Stirling numbers of the first and second kinds. As to umbral calculus, we give very basic facts such as Sheffer sequence, generating functions of Sheffer polynomials, and the formula for representing one Sheffer polynomial by another. For further details on umbral calculus, we let the reader refer to [3][4][5]. In Section 2, we find an explicit expression for the type 2 poly-Bernoulli polynomials involving Bernoulli numbers and Stirling numbers of the first kind, a recurrence relation for them, and an identity involving the type 2 poly-Bernoulli numbers and Stirling numbers of the first kind. In addition, we express the type 2 poly-Bernoulli polynomials as linear combinations of higher-order Cauchy polynomials, higher-order Euler polynomials, and of higher-order Frobenius-Euler polynomials.
It is one of our future projects to continue to work on various special polynomials and numbers by using umbral calculus, just as we did in the present paper.
Hardy introduced the polyexponential functions [6,7], while Kim-Kim considered the modified polyexponential functions which are given by From (1), we note that The type 2 poly-Bernoulli polynomials, which are defined by using the modified polyexponential functions, are given by For r ∈ , the Euler polynomials of order r are defined by e e E x t n For n 0 ≥ , the falling factorial sequence is defined by Here we note that the Stirling numbers of the first kind are defined by As an inversion formula of (7), the Stirling numbers of the second kind are defined by Let be the field of complex numbers and let be the algebra of formal power series. For , let * denote the vector space of all linear functional on . L p x | ( ) ⟨ ⟩denotes the action of the linear functional L on the polynomial p x ( ), and it is well known that the vector space operations on * are defined by where c is a complex constant (see [3][4][5]).
From (10), we note that where f t ( ) ∈ F and p x ( ) ∈ . Thus, by (11), we get where p x ( ) ∈ . Suppose that f t ( ) is a delta series and g t ( ) is an invertible series. Then there exists a unique sequence s x n ( ) of polynomials such that g t f t s x n δ k n n k , and Thus, by (14), we easily get For s x g t t , n ( ) ( ( ) ), we have and s x g t x n 1 , 0 s e e 1 5 .
We recall here that s x n ( ) is called the Appell sequence for g t ( ) if s x g t t , n ( ) ( ( ) ). For example, the sequence B x By ( where we used the trinomial coefficients Therefore, by (26), we obtain the following theorem.
where g t e t g t t g t 1 Ei log 1 and d d .
Therefore, we obtain the following theorem.
On the other hand,   For the next result, we recall that, for any r ∈ , the Cauchy polynomials C x n r ( ) ( ) of order r are given by We consider the following two Sheffer sequences.