Higher-order frobenius-Euler and poly-Bernoulli mixed type polynomials

In this paper, we consider higher-order Frobenius-Euler polynomi- als associated with poly-Bernoulli polynomials which are derived from polylogarithmic function. These polynomials are called higher-order Frobenius-Euler and poly-Bernoulli mixed type polynomials. The purpose of this paper is to give various identities of those polynomials arising from umbral calculus.


Introduction
For λ ∈ C with λ = 1, the Frobenius-Euler polynomials of order α(α ∈ R) are defined by the generating function to be t n n! , (see [1,6,7,13,14]). n (x) t n n! , (see [4,5,9] n (x) is called the n-th Bernoulli number of order α. In the special case, α = 1, B (1) n (x) = B n (x) is called the n-th Bernoulli polynomial. When x = 0, B n = B n (0) is called the n-th ordinary Bernoulli number. Finally, we recall that the Euler polynomials of order α are given by n (x) t n n! , (see [2,3,8,10,15] n (0) is called the n-th Euler number of order α. In the special case, α = 1, E (1) n (x) = E n (x) is called the n-th ordinary Euler polynomial. The classical polylogarithmic function Li k (x) is defined by x n n k , (k ∈ Z), (see [5]). (1.4) As is known, poly-Bernoulli polynomials are defined by the generating function to be t n n! , (cf. [5]). (1.5) Let C be the complex number field and let F be the set of all formal power series in the variable t over C with Now, we use the notation P = C[x]. In this paper, P * will be denoted by the vector space of all linear functionals on P. Let us assume that L | p(x) be the action of the linear functional L on the polynomial p(x), and we remind that the vector space operations on P * are defined by The formal power series defines a linear functional on P by setting f (t)|x n = a n , f or all n ≥ 0, (see [11,12]). (1.8) From (1.7) and (1.8), we note that t k |x n = n!δ n,k , (see [11,12]), (1.9) where δ n,k is the Kronecker symbol.
Let us consider f L (t) = ∞ k=0 L|x n k! t k . Then we see that f L (t)|x n = L|x n and so L = f L (t) as linear functionals. The map L → f L (t) is a vector space isomorphism from P * onto F . Henceforth, F will denote both the algebra of formal power series in t and the vector space of all linear functionals on P, and so an element f (t) of F will be thought of as both a formal power series and a linear functional (see [11]). We shall call F the umbral algebra. The umbral calculus is the study of umbral algebra. The order o(f (t)) of a nonzero power series f (t) is the smallest integer k for which the coefficient of t k does not vanish. A series f (t) is called a delta series if o(f (t)) = 1, and an invertible seires if o(f (t)) = 0. Let f (t), g(t) ∈ F . Then we have [11]). (1.10) For f (t), g(t) ∈ F with o(f (t)) = 1, o(g(t)) = 0, there exists a unigue sequence S n (x)(deg S n (x) = n) such that g(t)f (t) k |S n (x) = n!δ n,k f or n, k ≥ 0. The sequence S n (x) is called the Sheffer sequence for (g(t), f (t)) which is denoted by S n (x) ∼ (g(t), f (t)), (see [11,12]). Let f (t) ∈ F and p(t) ∈ P. Then we have (1.11) From (1.11), we note that (1.12) By (1.12), we get [11,12] (1.14) For p(x) ∈ P, f (t) ∈ F , it is known that , f (t)). Then we have , (see [11,12]). (1.17) The Stirling number of the second kind is defined by the generating function to be )S n (x), (n ≥ 0), (see [11,12]). [11,12]). (1.21) In this paper, we study higher-order Frobeniuns-Euler polynomials associated with poly-Bernoulli polynomials which are called higher-order Frobenius-Euler and poly-Beroulli mixed type polynomials. The purpose of this paper is to give various identities of those polynomials arising from umbral calculus.
is called the n-th higher-order Frobenius-Euler and poly-Bernoulli mixed type number.
Theorem 2.5. For r, k ∈ Z, n ≥ 1, we have On the one hand, 1 − e −t |x m On the other hand, we get n−l (λ).
Theorem 2.8. For r, k ∈ Z, s ∈ Z ≥0 , we have It is known that , t ,  Finally, we consider the following two Sheffer sequences: , t , x