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

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1 Adv. Studies Theor. Phys., Vol. 8, 04, no. 6, HIKARI Ltd, Symmetric Properties for the h, q-tangent Polynomials C. S. Ryoo Department of Mathematics Hannam University, Daeeon , Korea Copyright c 04 C. S. Ryoo. This is an open access article distributed under the 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. Mathematics Subect Classification: B68, S40, S80 Keywords: tangent numbers and polynomials, h, q-tangent polynomials, alternating sums, symmetric properties 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, 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 νpp = p. 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 <. If q C p, we normally assume that q p <p p so that q x = expx log q for x p. For g UD ={g g : C p is uniformly differentiable function},

2 60 C. S. Ryoo the fermionic p-adic invariant integral on is defined by Kim as follows: I g = gxdμ x = lim N p N x=0 If we take g x =gx + in., then we see that gx x, see[].. I g +I g =g0, see [].. 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: I q hy e yt = q hy e yt dμ y = I q hy e y+xt = q hy e y+xt dμ y = T n,q h t n n!,.3 T h n,q xtn n!..4 The following elementary properties of the h, q- tangent numbers E n,q h and polynomials T n,q h x are readily derived form.,.,.3 and.4 see, for details, [4]. We, therefore, choose to omit details involved. Theorem. For h Z, we have q hx x n dμ x =T n,q h, q hy y + x n dμ y =T n,q h x. Theorem. For any positive integer n, we have T h n,q x = n k=0 n T h k,q k xn k. The alternating sums of powers of consecutive h, q-even integers In this section, we assume that q C, with q < and h Z. By using.4, we give the alternating sums of powers of consecutive h, q-even integers as follows: T n,q h t n n! = q h e t + = n q nh e nt.

3 Symmetric properties for the h, q-tangent polynomials 6 From the above, we obtain n q nh e n+kt + k n k q n kh e nt = n k q n kh e nt. By using.3and.4, we obtain T h,q kt! + k q kh By comparing coefficients of t! T h t,q! = k k q kh in the above equation, we obtain k n q nh n = k+ q kh T h h,q k+t,q. By using the above equation we arrive at the following theorem: n q nh n Theorem. Let k be a positive integer and q C with q <. Then we obtain k T h,q k = n q nh n = k+ q kh T h h,q k+t,q.. Remark. For the alternating sums of powers of consecutive even integers, we have lim T h q k,q k = n n = k+ T k+t where T x and T 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 for Bernoulli polynomials. By using same method of [3], expect for obvious modifications, we investigate interesting properties of symmetry p-adic invariant integral on for h, q-tangent polynomials. By using., we have n I g n + n I g = n k gk, k=0, t!.

4 6 C. S. Ryoo where n N,g n x =gx + n. If n is odd from the above, we obtain n gx + ndμ x+ gxdμ x = n k gk. 3. k=0 Substituting gx =q hx e xt into the above, we obtain n q hx+n e x+nt dμ x+ q hx e xt dμ x = q h e t. 3. After some elementary calculations, we have q hx+n e x+nt dμ x+ q hx e xt dμ x = + qhn e nt. q h e t + By substituting Taylor series of e xt into 3. and the above, we arrive at the following theorem: Theorem 3. Let n be odd positive integer. Then we obtain q hx e xt dμ x q hnx e ntx dμ x = T h n t m. 3.3 Let w and w be odd positive integers. By using 3.3, we have q hw x +w x e w x +w x +w w xt dμ x dμ x q hw w x e w w xt dμ x = ew w xt q hw w e w w t + q hw e w t + q hw e w t By using 3.3 and 3.4, after elementary calculations, we obtain q hw x e w x +w w xt Z dμ x p q hw x e x w t dμ x q hw w x e w w tx dμ x = T h w w xw m t m T h w w w m t m. 3.5 By using Cauchy product in the above, we have m m T h,q w w xw T h m,q w w w m t m 3.6

5 Symmetric properties for the h, q-tangent polynomials 63 By using the symmetry in 3.5, we have q hw x e w x +w w xt Z dμ x p q hw x e x w t dμ x q hw w x e w w tx dμ x = T h w w xw m t m T h w w w m t m. Thus we obtain m m T h,q w w xw T h m,q w w w m t m 3.7 By comparing coefficients tm in the both sides of 3.6 and 3.7, we arrive at the following theorem: Theorem 3. Let w and w be odd positive integers. Then we have m m T h,q w w xt h m,q w w w m w m m = T h,q w w xt h m,q w w ww m, where T h h k,q x and T k denote the h, q-tangent polynomials and the alternating sums of powers of consecutive h, q-even integers, respectively. By using Theorem., we have the following corollary: Corollary 3.3 Let w and w be odd positive integers. Then we obtain m m w m k w k x k T h k,q w T h m,q w w k=0 = m k=0 m k w wm k x k T h k,q w T h m,q w w. By using 3.5, we have w ew w xt q hw x e x w t dμ x q wh e w t 0 x +w x+w A w t = q w h q hw x e w dμ x w = q wh T h n,q w w x + w w n t n w n!. 3.8

6 64 C. S. Ryoo By using the symmetry property in 3.8, we also have w ew w xt q hw x e x w t dμ x q wh e w t 0 x +w x+w A w t = q w h q hw x e w dμ x w = q wh T h n,q w w x + w w n t n w n!. 3.9 By comparing coefficients tn n! following theorem. in the both sides of 3.8 and 3.9, we have the Theorem 3.4 Let w and w be odd positive integers. Then we obtain w w = q w h T h n,q w q w h T h n,q w w x + w w n w w x + w w n. w 3.0 Observe that if q, then 3.0 reduces to Theorem 3.4 in [5]. Substituting w = into 3.0, we arrive at the following corollary. Corollary 3.5 Let w be odd positive integer. Then we obtain References w T n,q h x =wn q h T h n,q w x + [] L. Comtet, Advances combinatorics, Riedel, Dordrecht, 974. [] T. Kim, q-volkenborn integration, Russ. J. Math. Phys. 900, [3] T. Kim, On the Symmetries of the q-bernoulli Polynomials, Abstract and Applied Analysis, , Article ID 94367, 6 pages. [4] C. S. Ryoo, On the h, q-tangent Numbers and Polynomials, Adv. Studies Theor. Phys. 7903, w.

7 Symmetric properties for the h, q-tangent polynomials 65 [5] C. S. Ryoo, A Note on the Symmetric Properties for the Tangent Polynomials, Int. Journal of Math. Analysis, 7503, [6] H. Shin, J. Zeng, The q-tangent and q-secant numbers via continued fractions, European J. Combin. 300, Received: January 9, 04

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