Poly-Cauchy numbers and polynomials of the second kind

In this paper, we consider the poly-cauchy polynomials and numbers of the second kind which were studied by Komatsu in [10]. We note that the poly-Cauchy polynomials of the second kind are the special generalized Bernoulli polynomials of the second kind. The purpose of this paper is to give various identities of the poly-Cauchy polynomials of the second kind which are derived from umbral calculus.


Introduction
As is well known, the Bernoulli polynomials of the second kind are defined by the generating function to be t log (1 + t) (1 + t) x = ∞ n=0 b n (x) t n n! , (see [12]).
When x = 0, b n = b n (0) are called the Bernoulli numbers of the second kind. Let Lif k (x) be the polylogarithm factorial function which is defined by x m m!(m + 1) k , (see [9 − 11]). ( The poly-Cauchy polynomials of the second kindC (k) n (x) (k ∈ Z, n ∈ Z ≥0 ) are defined by the generating function to be n (x) t n n! , (see [10]).
Let C be the complex number field and let F be the set of all formal power series in the variable t: Let P = C[x] and let P * be the vector space of all linear functionals on P.
L|p(x) is 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 For f (t) ∈ F , let us define the linear functional on P by setting f (t)|x n = a n , (n ≥ 0).
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 (t)) of a power series f (t)( = 0) is the smallest integer k for which the coefficient of t k does not vanish. If O (f (t)) = 1, then f (t) is called a delta series ; if O (f (t)) = 0, then f (t) is called an invertible series (see [2,12,13]). For f (t), g(t) ∈ F with O (f (t)) = 1 and O (g(t)) = 0, 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 ≥ 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 [12,13]). For f (t), g(t) ∈ F and p(x) ∈ P, we have and Thus, by (12), we get Let us assume that s n (x) ∼ (g(t), f (t)). Then the generating function of s n (x) is given by wheref (t) is the compositional inverse of f (t) withf (f (t)) = t (see [12,13]). For s n (x) ∼ (g(t), f (t)), we have the following equation: and where p n (x) = g(t)s n (x), (see [12,13]). Let us assume that ). Then the transfer formula is given by , let us assume that Then we have [12]).
In this paper, we investigate the properties of the poly-Cauchy numbers and polynomials of the second kind with umbral calculus viewpoint. The purpose of this paper is to give various identities of the poly-Cauchy polynomials of the second kind which are derived from umbral calculus.

It is not difficult to show thatC
. Therefore, by (42), we obtain the following theorem.
In particular, if we take m = 1, then we get Remark. For s n (x) ∼ (g(t), f (t)), it is known that By (20) and (47), we easily show that d dxC Let us consider the following two Sheffer sequences: and Suppose thatC By (19), we see that Therefore, by (49) and (50), we obtain the following theorem. Remark. The Narumi polynomials of order a are defined by the generating function to be where, by (19), we get C n,m = 1 m!(1 − λ) r Lif k (− log (1 + t)) (1 + t − λ) r (log (1 + t)) m x n (57) Thus, by (57) and (58), we get Therefore, by (56) and (59), we obtain the following theorem. Therefore, by (60) and (61), we obtain the following theorem.