Some identities of higher-order Bernoulli, Euler and Hermite polynomials arising from umbral calculus

In this paper, we study umbral calculus to have alternative ways of obtaining our results. That is, we derive some interesting identities of the higher-order Bernoulli, Euler and Hermite polynomials arising from umbral calculus to have alternative ways.


Introduction
As is well known, the Hermite polynomials are defined by the generating function to be For λ ( = ) ∈ C, the Frobenius-Euler polynomials are given by In the special case, x = , H (r) n (|λ) = H (r) n (λ) are called the nth Frobenius-Euler numbers of order r.
Let F be the set of all formal power series in the variable t over C with Let us assume that P is the algebra of polynomials in the variable x over C and that P * is the vector space of all linear functionals on P. L|p(x) denotes the action of the linear functional L on a polynomial p(x), and we remind that the vector space structure on P * is defined by k! t k , we have f L (t)|x n = L|x n . 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 will 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. A series f (t) having o(f (t)) =  is called a delta series, and a series f (t) having o(f (t)) =  is called an invertible series. Let f (t) be a delta series and let g(t) be an invertible series. 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 , where n, k ≥ . The sequence S n (x) is called a Sheffer http://www.journalofinequalitiesandapplications.com/content/2013/1/211 sequence for (g(t), f (t)), which is denoted by S n (x) ∼ (g(t), f (t)). By (.) and (.), we see that e yt |p(x) = p(y). For f (t) ∈ F and p(x) ∈ P, we have and, by (.), we get Thus, from (.), we have Then we have Equations (.) and (.) are called the alternative ways of Sheffer sequences.
In this paper, we study umbral calculus to have alternative ways of obtaining our results. That is, we derive some interesting identities of the higher-order Bernoulli, Euler, and Hermite polynomials arising from umbral calculus to have alternative ways.

Some identities of higher-order Bernoulli, Euler, and Hermite polynomials
In this section, we use umbral calculus to have alternative ways of obtaining our results. Let us consider the following Sheffer sequences: Then, by (.), we assume that Therefore, by (.) and (.), we obtain the following theorem.
Theorem . For n ≥ , we have E (r) n (x) = n! n k= ≤l≤n-k,l:even Let us consider the following two Sheffer sequences: Let us assume that By (.) and (.), we get n-k-l = n! ≤l≤n-k,l:even Therefore, by (.) and (.), we obtain the following theorem.
Theorem . For n ≥ , we have B (r) n (x) = n! n k= ≤l≤n-k,l:even Let us assume that By (.), we get Therefore, by (.) and (.), we obtain the following theorem.
Let us assume that r ≤ n. For  ≤ k < r, by (.), we get For r ≤ k ≤ n, by (.), we get