Symmetric Identities of Hermite-Bernoulli Polynomials and Hermite-Bernoulli Numbers Attached to a Dirichlet Character χ

We aim to introduce arbitrary complex order Hermite-Bernoulli polynomials and Hermite-Bernoulli numbers attached to a Dirichlet character χ and investigate certain symmetric identities involving the polynomials, by mainly using the theory of p-adic integral on Zp. The results presented here, being very general, are shown to reduce to yield symmetric identities for many relatively simple polynomials and numbers and some corresponding known symmetric identities.

numbers attached to a Dirichlet character χ

Introduction and Preliminaries
For a fixed prime number p, throughout this paper, let Z p , Q p , and C p be the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Q p , respectively.In addition, let C, Z, and N be the field of complex numbers, the ring of rational integers and the set of positive integers, respectively, and let N 0 := N ∪ {0}.Let UD(Z p ) be the space of all uniformly differentiable functions on Z p .The notation [z] q is defined by [z] q := 1 − q z 1 − q (z ∈ C; q ∈ C \ {1}; q z = 1) .
We aim to introduce arbitrary complex order Hermite-Bernoulli polynomials attached to a Dirichlet character χ and investigate certain symmetric identities involving the polynomials (15) and (31), by mainly using the theory of p-adic integral on Z p .The results presented here, being very general, are shown to reduce to yield symmetric identities for many relatively simple polynomials and numbers and some corresponding known symmetric identities.
Using Equation ( 4), we have Z p e η 2 t u du Using Equations ( 5) and ( 15), we find Employing a formal manipulation of double series (see, e.g., [32] (Equation (1.1))) with p = 1 in the last two series in Equation (20), and again, the resulting series and the first series in Equation ( 20), we obtain Noting the symmetry of F(α; η 1 , η 2 )(t) with respect to the parameters η 1 and η 2 , we also get Equating the coefficients of t n in the right sides of Equations ( 22) and ( 23), we obtain the first equality of Equation (16).
For (17), we write Noting we have Using Equation ( 15), we obtain Applying Equation ( 21) with p = 1 to the right side of Equation ( 26), we get In view of symmetry of F(α; η 1 , η 2 )(t) with respect to the parameters η 1 and η 2 , we also obtain Equating the coefficients of t n in the right sides of Equation (27) and Equation ( 28), we have Equation (17).

Symmetry Identities of Arbitrary Order Hermite-Bernoulli Polynomials Attached to a Dirichlet Character χ
We begin by introducing generalized Hermite-Bernoulli polynomials attached to a Dirichlet character χ of order α ∈ C defined by means of the following generating function: where χ is a Dirichlet character with conductor d.
and n ∑ m=0 where χ is a Dirichlet character with conductor d.

Conclusions
The results in Theorems 1 and 2, being very general, can reduce to yield many symmetry identities associated with relatively simple polynomials and numbers using Remarks 1-5.Setting z = 0 and α ∈ N in the results in Theorem 1 and Theorem 2 yields the corresponding known identities in References [33,34], respectively.