Symmetric Properties for the (h, Q)-tangent Polynomials

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In [4], we studied the (h, q)-tangent numbers and polynomials. By using these numbers and polynomials, we give some interesting symmetric properties for the (h, q)-tangent polynomials.


Introduction
Throughout this paper, we always make use of the following notations: N denotes the set of natural numbers and Z + = N ∪ {0} , C denotes the set of complex numbers, Z p denotes the ring of p-adic rational integers, Q p denotes the field of p-adic rational numbers, and C p denotes the completion of algebraic closure of Q p .Let ν p be the normalized exponential valuation of C p with |p| p = p −νp(p) = p −1 .When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ C p .If q ∈ C one normally assume that |q| < 1.If q ∈ C p , we normally assume that |q − 1| p < p − 1 p−1 so that q x = exp(x log q) for |x| p ≤ 1.For the fermionic p-adic invariant integral on Z p is defined by Kim as follows: x , (see [2]). (1.1) If we take g 1 (x) = g(x + 1) in (1.1), then we see that In [4], we introduced the (h, q)-tangent numbers T (h) n,q and polynomials T (h) n,q (x) and investigate their properties.Let us define the (h, q)-tangent numbers T (h) n,q and polynomials T (h) n,q (x) as follows: The following elementary properties of the (h, q)-tangent numbers and polynomials T (h) n,q (x) are readily derived form (1.1), (1.2), (1.3) and (1.4)( see, for details, [4]).We, therefore, choose to omit details involved.Theorem 1.1 For h ∈ Z, we have p q hx (2x) n dμ −1 (x) = T (h)  n,q , p q hy (2y + x) n dμ −1 (y) = T (h) n,q (x).
Theorem 1.2 For any positive integer n, we have

2
The alternating sums of powers of consecutive (h, q)-even integers In this section, we assume that q ∈ C, with |q| < 1 and h ∈ Z.By using (1.4), we give the alternating sums of powers of consecutive (h, q)-even integers as follows: From the above, we obtain By using (1.3)and (1.4), we obtain By comparing coefficients of t j j! in the above equation, we obtain By using the above equation we arrive at the following theorem: Theorem 2.1 Let k be a positive integer and q ∈ C with |q| < 1.Then we obtain

Remark 2.2 For the alternating sums of powers of consecutive even integers, we have
where T j (x) and T j denote the tangent polynomials and the tangent numbers, respectively.
3 Symmetric properties for the (h, q)-tangent polynomials In this section, we assume that q ∈ C p and ∈ T p .In [3], Kim investigated interesting properties of symmetry p-adic invariant integral on Z p for Bernoulli polynomials.By using same method of [3], expect for obvious modifications, we investigate interesting properties of symmetry p-adic invariant integral on Z p for (h, q)-tangent polynomials.By using (1.1), we have where n ∈ N, g n (x) = g(x + n).If n is odd from the above, we obtain Substituting g(x) = q hx e 2xt into the above, we obtain After some elementary calculations, we have p q h(x+n) e (2x+2n)t dμ −1 (x) + p q hx e 2xt dμ −1 (x) = 2(1 + q hn e 2nt ) q h e 2t + 1 .
By substituting Taylor series of e 2xt into (3.2) and the above, we arrive at the following theorem: Theorem 3.1 Let n be odd positive integer.Then we obtain Let w 1 and w 2 be odd positive integers.By using (3.3), we have p p q h(w 1 x 1 +w 2 x 2 ) e (w 1 2x 1 +w 2 2x 2 +w 1 w 2 x)t dμ −1 (x 1 )dμ −1 (x 2 ) p q hw 1 w 2 x e 2w 1 w 2 xt dμ −1 (x) = 2e w 1 w 2 xt (q hw 1 w 2 e 2w 1 w 2 t + 1) (q hw 1 e 2w 1 t + 1)(q hw 2 e 2w 2 t + 1) (3.4)By using (3.3) and (3.4), after elementary calculations, we obtain (3.5)By using Cauchy product in the above, we have By using the symmetry in (3.5), we have Thus we obtain By comparing coefficients t m m! in the both sides of (3.6) and (3.7), we arrive at the following theorem: Theorem 3.2 Let w 1 and w 2 be odd positive integers.Then we have where m,q (k) denote the (h, q)-tangent polynomials and the alternating sums of powers of consecutive (h, q)-even integers, respectively.By using Theorem 1.2, we have the following corollary: m−j,q w 2 (w 1 − 1).

Corollary 3 . 3
Let w 1 and w 2 be odd positive integers.Then we obtain