Sheffer sequences of polynomials and their applications

In this paper, we investigate some properties of several Sheffer sequences of several polynomials arising from umbral calculus. From our investigation, we can derive many interesting identities of several polynomials


Introduction
As is well known, the Bernoulli polynomials of order a are defined by the generating function to be Let Let P be the algebra of polynomials in the variable x over C and P * be the vector space of all linear functionals on P. The action of the linear functional L on a polynomial p(x) is denoted by L|p(x) . We recall that the vector space structures on P * are defined by By (.) and (.), we get Then we have f L (t)|x n = L|x n and f L (t) = L. Thus, we note that the map L → f L (t) is a vector space isomorphism from P * onto F . Henceforth, F will be thought of as both a formal power series and a linear functional. We shall call F the umbral algebra. The umbral calculus is the study of umbral algebra (see [-]).
The order o(f (t)) of the non-zero power series f (t) is the smallest integer k for which the coefficient of t k does not vanish. If o(f (t)) = , then f (t) is called a delta series. If o(f (t)) = , then f (t) is called an invertible series. Let o(f (t)) =  and o(g(t)) = . Then there exists a unique sequence S n (x) of polynomials such that g(t)f (t) k |S n (x) = n!δ n,k (n, k ≥ ). The sequence S n (x) is called Sheffer sequence for (g(t), f (t)), which is denoted by S n (x) ∼ (g(t), f (t)). By (.), we easily get that e yt |p(x) = p(y). For f (t) ∈ F and p(x) ∈ P, we have Thus, by (.), we have (.) http://www.advancesindifferenceequations.com/content/2013/1/118 . For a = , the Poisson-Charlier sequences are given by The Frobenius-type Eulerian polynomials of order a are given by From (.) and (.), we note that Let us assume that p n (x) ∼ (, f (t)), q n (x) ∼ (, g(t)). Then we have Equation (.) is important in deriving our results in this paper. The purpose of this paper is to investigate some properties of Sheffer sequences of several polynomials arising from umbral calculus. From our investigation, we can derive many interesting identities of several polynomials.

Sheffer sequences of polynomials
Let us assume that S n (x) ∼ (g(t), f (t)). Then, by the definition of Sheffer sequence, we see that is an invertible series, then  g(t) is also an invertible series. Let us consider the following Sheffer sequences: x n ∼ (, t).
Theorem . For S n (x) ∼ (g(t), f (t)) and n ≥ , we have , where D n (x) is the nth Daehee polynomial (see [, , ]). Then, by Theorem ., we get Let us take the following Sheffer sequence: By Theorem . and (.), we get where E (α) n (x) are the nth Euler polynomials of order α which is defined by the generating function to be Therefore, by (.), we obtain the following theorem.
Theorem . For n ≥ , let S n (x) ∼ (( e t +  ) α , t  log(+t) ). Then we have As is known, we note that Corollary . For n ≥ , and  ≤ l ≤ n -, we have . Then, by Theorem ., we get Let us assume that Then, by Theorem . and (.), we get where n ≥ ,  ≤ l ≤ n - and c = . Let From Theorem . and (.), we note that Therefore, by (.), we obtain the following proposition.