Some properties of higher-order Daehee polynomials of the second kind arising from umbral calculus

In this paper, we study the higher-order Daehee polynomials of the second kind from the umbral calculus viewpoint and give various identities of the higher-order Daehee polynomials of the second kind arising from umbral calculus.


Introduction
Let k ∈ Z ≥ . The Daehee polynomials of the second kind of order k are defined by the generating function to be As is well known, the Bernoulli polynomials of order k (∈ N) are defined by the generating function to be . When x = , B (k) n = B (k) n () are called the Bernoulli numbers of order k. In this paper, we study the higher-order Daehee polynomials of the second kind with umbral calculus viewpoint and give various identities of the higher-order Daehee polynomials of the second kind arising from umbral calculus.

Umbral calculus
Let C be the complex number field and let F be the set of all formal power series Let P = C[x], and let P * be the vector space of all linear functionals on P. L|p(x) indicates the action of the linear functional L on the polynomial p(x). Then the vector space operations on P * are given by where c is a complex constant in C. For f (t) ∈ F , the linear functional on P is defined by f (t)|x n = a n . Then, in particular, we have The map L → f L (t) is a vector space isomorphism from P * onto F . Henceforth, F denotes both the algebra of the 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 the umbral algebra. The order o(f (t)) of the power series f (t) ( = ) is the smallest integer for which the coefficient of t k does not vanish. If o(f (t)) = , then f (t) is called an invertible series; if o(f (t)) = , then f (t) is called a delta series.
Let f (t), g(t) ∈ F with o(f (t)) =  and o(g(t)) = . Then there exists a unique sequence s n (x) (deg s n (x) = n) such that g(t)f (t) k |s n (x) = n!δ n,k , for n, k ≥ . The sequence s n (x) is called the Sheffer sequence for (g(t), f (t)) which is denoted by From (), we note that and, by (), we get Let us assume that s n (x) ∼ (g(t), f (t)) and r n (x) ∼ (h(t), l(t)). Then we see that where

Higher-order Daehee polynomials of the second kind
By (), we see that From () and (), we havê Therefore, by () and (), we obtain the following theorem.
Theorem  For n ∈ Z ≥ and k ≥ , we havê By () and (), we get Therefore, by (), we obtain the following theorem. and Thus, by (), we get From () and (), we derive the following equation: Therefore, from () and (), we obtain the following theorem.
Theorem  For n ≥ , k ≥ , we havê Now, we observe that e -t e t -t t(e t -)D Thus, by (), we get From () and (), we note that d dxD By () and (), we see that Thus, by (), we get Therefore, by (), we obtain the following theorem.
Theorem  For n ≥ , k ≥ , we havê Now, we compute ( (+t) log(+t) t ) k (log( + t)) m |x n in two different ways: On the other hand, Therefore, by () and (), we obtain the following theorem.