Hermite and poly-Bernoulli mixed-type polynomials

In this paper, we consider Hermite and poly-Bernoulli mixed-type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities associated with Stirling numbers, Bernoulli and Frobenius-Euler polynomials of higher order.

For k ∈ Z, the polylogarithm is defined by x n n k . t n n! , (see [5,8]) .
n (0) are called the Hermite numbers of order ν. In this paper, we consider the Hermite and poly-Bernoulli mixed-type polynomials HB (ν,k) n (x) which are defined by the generating function to be where k ∈ Z and ν ( = 0) ∈ R. When x = 0, HB (ν,k) n = HB (ν,k) n (0) are called the Hermite and poly-Bernoulli mixed-type numbers.
Let F be the set of all formal power series in the variable t over C as follows: Let P = C [x] and P * denote the vector space of all linear functionals on P.
L| p (x) denotes the action of the linear functional L on the polynomial p (x), and we recall that the vector space operations on P * are defined by where c is complex constant in C. For f (t) ∈ F , let us define the linear functional on P by setting (1.7) f (t)| x n = a n , (n ≥ 0) .
The map L −→ f L (t) is a vector space isomorphism from P * onto F . Henceforth, F denotes 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. We call F the umbral algebra and the umbral calculus is the study of umbral algebra. The order O (f ) of the power series f (t) = 0 is the smallest integer for which a k does not vanish. If Let f (t) ∈ F and p (x) ∈ P. Then we have (1.10) [8,9,11,13,14]) .
In this paper, we consider Hermite and poly-Bernoulli mixed-type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities associated with Bernoulli and Frobenius-Euler polynomials of higher order.
From (1.5), we can also derive Therefore, by (2.6), we obtain the following theorem.
Theorem 3. For n ≥ 0, we have By (2.6), we get where S 2 (n, m) is the Stirling number of the second kind. Therefore, by (2.8), we obtain the following theorem.
Therefore, by (2.9), we obtain the following theorem.
By (2.12) and (2.13), we get It is easy to show that Thus, by (2.15), we get
Theorem 6. For n ≥ 0, we have Let us take t on the both sides of (2.18). Then we have Now, we observe that From (2.21), we have in two different ways.
On the one hand, Therefore, by (2.23) and (2.24), we obtain the following theorem.
Theorem 9. For n ≥ 0, we have