SS symmetry Article
Symmetric Identities of Hermite-Bernoulli Polynomials and Hermite-Bernoulli Numbers Attached to a Dirichlet Character χ Serkan Araci 1, * 1 2 3
*
ID
, Waseem Ahmad Khan 2 and Kottakkaran Sooppy Nisar 3
ID
Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey Department of Mathematics, Faculty of Science, Integral University, Lucknow-226026, India;
[email protected] Department of Mathematics, College of Arts and Science-Wadi Aldawaser, Prince Sattam bin Abdulaziz University, 11991 Riyadh Region, Kingdom of Saudi Arabia;
[email protected] or
[email protected] Correspondence:
[email protected]
Received: 14 November 2018; Accepted: 27 November 2018; Published: 29 November 2018
Abstract: 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 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. Keywords: q-Volkenborn integral on Z p ; Bernoulli numbers and polynomials; generalized Bernoulli polynomials and numbers of arbitrary complex order; generalized Bernoulli polynomials and numbers attached to a Dirichlet character χ
1. 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 N0 := N ∪ {0}. Let UD(Z p ) be the space of all uniformly differentiable functions on Z p . The notation [z]q is defined by 1 − qz 1−q
[z]q :=
( z ∈ C; q ∈ C \ { 1 } ; q z 6 = 1 ) .
Let νp be the normalized exponential valuation on C p with | p| p = pνp ( p) = p−1 . For f ∈ UD(Z p ) and q ∈ C p with |1 − q| p < 1, q-Volkenborn integral on Z p is defined by Kim [1] Iq ( f ) =
Z
f ( x ) dµq ( x ) = lim
N →∞
Zp
1
[ p N ]q
p N −1
∑
f (x) qx .
(1)
x =0
For recent works including q-Volkenborn integration see References [1–10]. The ordinary p-adic invariant integral on Z p is given by [7,8] I1 ( f ) = lim Iq ( f ) = q →1
Symmetry 2018, 10, 675; doi:10.3390/sym10120675
Z
f ( x ) dx.
(2)
Zp
www.mdpi.com/journal/symmetry
Symmetry 2018, 10, 675
2 of 10
It follows from Equation (2) that I1 ( f 1 ) = I1 ( f ) + f 0 (0),
(3)
where f n ( x ) := f ( x + n) (n ∈ N) and f 0 (0) is the usual derivative. From Equation (3), one has Z
∞ t tn = B n ∑ n! , et − 1 n =0
e xt dx = Zp
(4)
where Bn are the nth Bernoulli numbers (see References [11–14]; see also Reference [15] (Section 1.7)). From Equation (2) and (3), one gets n R
R
n −1
=
e xt dx
Z
1 = nxt dx t e Zp Zp
∑
jt
e =
j =0
e
( x +n)t
dx −
Z
Zp
∞
n −1
k =0
j =0
∑ ∑j
xt
e dx Zp
! k
(5)
∞ tk tk = ∑ Sk ( n − 1) , k! k! k =0
where Sk ( n ) = 1k + · · · + n k
(k ∈ N, n ∈ N0 ) .
(6)
(α)
From Equation (4), the generalized Bernoulli polynomials Bn ( x ) are defined by the following p-adic integral (see Reference [15] (Section 1.7)) Z
··· Zp | {z
Z
α times
e
( x +y1 +y2 +···+yα )t
dy1 dy2 · · · dyα =
Zp
t et − 1
α
e xt =
∞
∑ Bn
(α)
n =0
(x)
tn n!
(7)
}
(1)
in which Bn ( x ) := Bn ( x ) are classical Bernoulli numbers (see, e.g., [1–10]). Let d, p ∈ N be fixed with (d, p) = 1. For N ∈ N, we set N X = Xd = lim Z /dp Z ; ← − N
n o a + dp Z p = x ∈ X | x ≡ a mod dp N a ∈ Z with 0 ≤ a < dp N ; [ X∗ = a + dpZ p , X1 = Z p . N
(8)
0< a
