Some Identities for the Higher-order q-Bernoulli Polynomials of the Second Kind under the Third Dihedral Group D 3

In this paper, we consider the higher-order q-Bernoulli polynomials of the second kind and investigate some symmetric identities under the third Dihedral group D3 which are derived from multivariate p-adic invariant integral on Zp.


Introduction
Let p be a fixed prime number.Throughout this paper, Z p , Q p , and C p will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of the algebraic closure of Q p .Let v p be the normalized exponential valuation of C p with |p| p = 1/p and let q be an indeterminate in C p with |1 − q| p < p −1/(p−1) .As is well known, the q-number of x is defined by [x] q = (1 − q x )/(1 − q).Note that lim q→1 [x] q = x.Let us assume that f (x) is uniformly differentiable function on Z p .The p-adic invariant integral on Z p is defined to be ˆZp f (x)dµ 0 (x) = lim [8,10,12]). (1) From (1), we have where where B n (x) are called Bernoulli polynomials.When x = 0, B n = B n (0) are called the Bernoulli numbers (see [8,9,10]).
For r ∈ N, we consider the higher-order q-Bernoulli polynomials of the second kind as follows : When x = 0, β n,q (0) are called the Bernoulli numbers of order r.Thus, by (7), we get In this paper, we investigate some symmetric identities for the higher-order q-Bernoulli polynomials of the second kind under the third Dihedral group D 3 .
2. Some identities for the higher-order q-Bernoulli polynomials of the second kind Then we see that ˆZp From ( 9), we can derive the following equation: By the same method as (10), we get where Therefore, by (10) and (11), we obtain the following theorem.
are the same for any σ ∈ D 3 , where By ( 8), we get Therefore, by Theorem 1 and ( 12), we obtain the following theorem.

Now, we observe that
From (13), we can derive the following equation : By ( 10), ( 12) and ( 14), we get k,q w 2 w 3 (w 1 x)T (r) n,q w 1 (w 2 , w 3 |k), where T (r) n,q (w 1 , w 2 |k) = As this expression is an invariant under third Dihedral group D 3 , we have the following theorem.