A Note on the Carlitz s Type Twisted q-tangent. Numbers and Polynomials

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1 Applied Mathematical Sciences, Vol. 12, 2018, no. 15, HIKARI Ltd A Note on the Carlitz s Type Twisted q-tangent Numbers and Polynomials Cheon Seoung Ryoo Department of Mathematics Hannam University, Daejeon , Korea Copyright c 2018 Cheon Seoung Ryoo. This article is 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 this paper we introduce the Carlitz s type twisted q-tangent numbers T n,q,ω and polynomials T n,q,ω (x. From these numbers and polynomials, we establish some interesting identities and relations. Mathematics Subject Classification: 11B68, 11S40, 11S80 Keywords: Tangent numbers and polynomials, q-tangent numbers and polynomials, twisted q-tangent numbers and polynomials, Carlitz s type twisted q-tangent numbers and polynomials 1 Introduction Mathematicians have worked in the area of the tangent numbers and polynomials(see [1-8]. In this paper, we define q-analogue of tangent polynomials and numbers and study some properties of the q-analogue of tangent numbers and polynomials. Throughout this paper, we always make use of the following notations: N denotes the set of natural numbers, Z + = N {0} denotes the set of nonnegative integers, and Z 0 = {0, 1, 2, 2,...} denotes the set of nonpositive integers, Z denotes the set of integers. Let r be a positive integer, and let ω be r-th root of unity. We remember that the classical twisted tangent numbers T n,ω and twisted tangent polynomials T n,ω (x are defined by the

2 732 Cheon Seoung Ryoo following generating functions(see [3] and ( 2 e xt = ωe 2t ωe 2t + 1 = t n T n,ω n!, (1.1 T n,ω (x tn n!. (1.2 respectively. Some interesting properties of the classical twisted tangent numbers and polynomials were first investigated by Ryoo [3]. Many kinds of of generalizations of these numbers and polynomials have been presented in the literature(see [ 2, 3, 4, 5, 6, 7]. The q-number is defined by [n] q = 1 qn 1 q. In particular, we can see lim q 1 [n] q = n. By using q-number, we construct the Carlitz s type twisted q-tangent numbers and polynomials, which generalized the previously known numbers and polynomials, including the twisted tangent numbers and polynomials. In the following section, we introduce the Carlitz s type twisted q-tangent numbers and polynomials. After that we will investigate some their properties. 2 Carlitz s type twisted q-tangent numbers and polynomials In this section, we define the Carlitz s type twisted q-tangent numbers and polynomials and provide some of their relevant properties. Let r be a positive integer, and let ω be r-th root of 1. Definition 2.1 For q < 1, the Carlitz s type twisted q-tangent numbers T n,q,ω and polynomials T n,q,ω (x are defined by means of the generating functions and respectively. F q,ω (t = F q,ω (t, x = T n,q,ω (x tn n! = [2] q T n,q,ω (x tn n! = [2] q ( 1 m ω m q m e [2m]qt. (2.1 ( 1 m ω m q m e [2m+x]qt, (2.2

3 On Carlitz s type twisted q-tangent numbers and polynomials 733 Specially, setting q 1 in (2.1 and (2.1, we obtain the corresponding definitions for the twisted tangent number T n,ω and tangent polynomials T n,ω (x respectively. By using above equation (2.1, we have t n T n,q,ω ( 1 m ω m q m e [2m]qt n! = [2] q ( = [2] q ( 1 1 q n n ( n ( 1 l 1 t n l 1 + ωq 2l+1 n!. (2.3 By comparing the coefficients tn n! in the above equation, we have the following theorem. Theorem 2.2 For n Z +, we have ( n n ( 1 n T n,q,ω = [2] q ( 1 l 1 1 q l 1 + ωq. 2l+1 Again, by (2.2, we obtain T n,q,ω (x = [2] q ( 1 1 q By using (2.2 and (2.4, we obtain ( T n,q,ω (x tn n! = ( 1 [2] q 1 q n n n n = [2] q ( 1 m ω m q m e [2m+x]qt. ( n ( 1 l q xl 1. (2.4 l 1 + ωq2l+1 ( n ( 1 l q xl 1 t n l 1 + ωq 2l+1 n! (2.5 Since [x + 2y] q = [x] q + q x [2y] q, we see that n ( n l ( ( l l 1 T n,q,ω (x = [2] q [x] n l q q xl ( 1 k 1 l k 1 q 1 + ωq. 2k+1 k=0 (2.6 By using (2.6 and Theorem 2.2, we have the following theorem. Theorem 2.3 For n Z +, we have n ( n T n,q,ω (x = [x] n l q q xl T l,q,ω l = (q x T q,ω + [x] q n = [2] q ( 1 m ω m q m [2m + x] n q, with the usual convention of replacing (T q,ω n by T n,q,ω.

4 734 Cheon Seoung Ryoo The following elementary properties of the Carlitz s type twisted q-tangent numbers T n,q,ω and polynomials T n,q,ω (x are readily derived form (2.1 and (2.2. We, therefore, choose to omit details involved. Theorem 2.4 (Distribution relation. For any positive integer m(=odd, we have, n Z +. T n,q,ω (x = [2] m 1 q [m] n q [2] q m a=0 ( 1 a ω a q a T n,q m,ω m ( 2a + x m Theorem 2.5 (Property of complement. By (2.1 and (2.2, we get [2] q Hence we have T n,q 1,ω 1(2 x = ( 1n ωq n T n,q,ω (x ( 1 l+n ω l+n q l+n e [2l+2n]qt + [2] q n 1 = [2] q ( 1 l ω l q l e [2l]qt. ( 1 l ω l q l e [2l]qt (2.7 ( 1 n+1 ω n q n T m,q,ω (2n tm m! + T m,q,ω (2n tm m! ( (2.8 n l = [2] q ( 1 l ω l q l [2l] m t m q m!. By comparing the coefficients tm m! on both sides of (2.8, we have the following theorem. Theorem 2.6 For n Z +, we have n l ( 1 l ω l q l [2l] m q = ( 1n+1 ω n q n T mq,ω (2n + T m,q,ω [2] q. 3 Carlitz s type twisted q-tangent zeta function By using Carlitz s type twisted q-tangent numbers and polynomials, Carlitz s type twisted q-tangent zeta function and Hurwitz twisted q-tangent zeta functions are defined. These functions interpolate the Carlitz s type twisted q- tangent numbers T n,q,ω, and polynomials T n,q,ω (x, respectively. From (2.1,

5 On Carlitz s type twisted q-tangent numbers and polynomials 735 we note that d k dt F q,ω(t k = [2] q t=0 ( 1 n ω m q m [2m] k q = T k,q,ω, (k N. By using the above equation, we are now ready to define the Carlitz s type twisted q-tangent zeta function. Definition 3.1 Let s C with Re(s > 0. ζ q,,ω (s = [2] q n=1 ( 1 n ω n q n. (3.1 [2n] s q Note that ζ q,ω (s is a meromorphic function on C. Note that, if q 1, then ζ q,ω (s = ζ ω (s which is the twisted tangent zeta functions(see [1]. Relation between ζ q,ω (s and T k,q,ω is given by the following theorem. Theorem 3.2 For k N, we have ζ p,q ( k = T k,p,q. Observe that ζ q,ω (s function interpolates T k,q,ω numbers at non-negative integers. From (2.2, we note that d k dt F q,ω(t, x k = [2] q ( 1 m ω m q m [2m + x] k q (3.2 t=0 and ( ( k d T n,q,ω (x tn = T dt n! k,q,ω (x, for k N. (3.3 t=0 By (3.2 and (3.3, we are now ready to define the Carlitz s type Hurwitz twisted q-tangent zeta function. Definition 3.3 Let s C with Re(s > 0 and x / Z 0. ζ q,ω (s, x = [2] q ( 1 n ω n q n. (3.4 [2n + x] s q Note that ζ q,ω (s, x is a meromorphic function on C. Obverse that, if q 1, then ζ q,ω (s, x = ζ ω (s, x which is the Hurwitz tangent zeta functions(see [1]. Relation between ζ q,ω (s, x and T k,q,ω (x is given by the following theorem. Theorem 3.4 For k N, we have ζ q,ω ( k, x = T k,q,ω (x. Observe that ζ q,ω ( k, x function interpolates T k,q,ω (x numbers at non-negative integers.

6 736 Cheon Seoung Ryoo 4 Carlitz s type twisted q-tangent numbers and polynomials associated with p-adic q- integral on Throughout this section we use the notation: 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 g UD( = {g g : C p is uniformly differentiable function}, Kim[1] defined the p-adic q-integral on as follows: 1 + q I q (g = g(xdµ q (x = lim Z N p 1 + q pn p N 1 x=0 g(x( q x. (4.1 From (4.1, we note that n 1 q n I q (f n = ( 1 n I q (g + [2] q ( 1 n 1 l q l g(l, (4.2 where g n (x = g(x + n for n N. Let T p = N 1 C p N = lim N C p N, where C p N = {ω ω pn = 1} is the cyclic group of order p N. For ω T p, we denote by φ ω : C p the locally constant function x ω x. By using p-adic q-integral on, we obtain, ω x [2x] n q dµ q (x = lim N 1 [p N ] q [2x] n q ω x ( q x p N 1 x=0 = [2] q ( 1 m ω m q m [2m] n q. (4.3 By using (2.1 and (4.3, we have t n T n,q,ω n! = ω Zp x [2x] nq dµ q (x tn n! = ω x e [2x]qt dµ q (x. (4.4

7 On Carlitz s type twisted q-tangent numbers and polynomials 737 Again, by using p-adic q-integral on, we obtain ω y [2y + x] n q dµ q (y = lim N 1 [p N ] q [2y + x] n q ω y ( q y p N 1 y=0 = [2] q ( 1 m ω m q m [2m + x] n q. (4.5 By (4.5 and (2.2, we obtain the following Witts formula. Theorem 4.1 For n Z, we have From (4.2, we note that T n,q,ω = ω x [2x] n q dµ q (x, T n,q,ω (x = ω y [2y + x] q dµ q (y. [2] q = qω ω x e [2x+2]qt dµ q (x + ω x e [2x]qt dµ q (x Z ( p = qω ω x [2x + 2] n q dµ q (x + ω x [2x] n q dµ q (x = (qwt n,q,ω (2 + T n,q,ω tn n!. Therefore, we obtain the following theorem. Theorem 4.2 For n N 0, we have 5 Conclusion qωt n,q,ω (2 + T n,q,ω = { [2]q, if n = 0, 0, if n 0. In this article, we defined the Carlitz s type twisted q-tangent numbers and polynomials. We have derived several formulas for Carlitz s type twisted q- tangent numbers and polynomials. In the limit when q 1, the results of this paper are the same as those of [3]. Also, if we take ω = 1, then [2] is the special case of this paper. t n n!

8 738 Cheon Seoung Ryoo References [1] T. Kim, q-euler numbers and polynomials associated with p-adic q- integrals, J. Nonlinear Math. Phys., 14 (2007, [2] K.W. Hwang, C.S. Ryoo, On Carlitz s type q-tangent numbers and polynomials and computation of their zeros, J. Appl. Math. & Informatics, 35 (2017, [3] C.S. Ryoo, On the analogues of tangent numbers and polynomials associated with p-adic integral on, Applied Mathematical Sciences, 7 (2013, [4] C.S. Ryoo, A numerical investigation on the zeros of the tangent polynomials, J. Appl. Math. & Informatics, 32 (2014, [5] C.S. Ryoo, A note on the tangent numbers and polynomials, Adv. Studies Theor. Phys., 7 (2013, [6] C.S. Ryoo, Differential equations associated with tangent numbers, J. Appl. Math. & Informatics, 34 (2016, [7] C.S. Ryoo, On degenerate q-tangent polynomials of higher order, J. Appl. Math. & Informatics, 35 (2017, [8] H. Shin, J. Zeng, The q-tangent and q-secant numbers via continued fractions, European J. Combin., 31 (2010, Received: June 11, 2018; Published: June 26, 2018