Some Properties Related to the Generalized q-genocchi Numbers and Polynomials with Weak Weight α

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1 Appied Mathematica Sciences, Vo. 6, 2012, no. 118, Some Properties Reated to the Generaized q-genocchi Numbers and Poynomias with Weak Weight α J. Y. Kang Department of Mathematics Hannam University, Daejeon , Korea C. S. Ryoo Department of Mathematics Hannam University, Daejeon , Korea Abstract Recenty many mathematicians are working on Genocchi numbers and Genocchi poynomias. We define a new generaized q-genocchi numbers n,χ,q and poynomias G n,χ,q(x (α with weak weight α and give some interesting reations of their numbers and poynomias with weak weight α. Aso, we construct generaized q-genocchi zeta function and generaized Hurwitz q-genocchi zeta function and find reations between generaized q-genocchi numbers and poynomias with weak weight α and their zeta functions. Mathematics Subject Cassification: 11B68, 11S40, 11S80 Keywords: generaized q-genocchi numbers and poynomias with weak weight α, generaized q-genocchi zeta function 1 Introduction Many mathematicians defined the generaized q-genocchi numbers and poynomias by using p-adic invariant integras on Z p (see [1-9]. Aso they introduced generaized q-genocchi zeta function which interpoate q-genocchi poynomias, in [1,3,4,6,9]. In the paper, our aim is to construct the new generaized q- Genocchi numbers n,χ,q and poynomias G n,χ,q(x (α with weak weight α by

2 5852 J. Y. Kang and C. S. Ryoo using q-vokenborn integration. Next we construct new generaized q-genocchi zeta function and new generaized Hurwitz q-genocchi function which interpoate the generaized q-genocchi numbers and poynomias with weak weight α at negative integer. Throughout this paper we use the foowing notations. Let p be a fixed odd prime number. Throughout this paper Z p, Q p, C, and C p denote the ring of p-adic rationa integers, the fied of p-adic rationa numbers, the compex number fied, and the competion of the agebraic cosure of Q p, respectivey. Let N be the set of natura numbers and Z + = N {0}. Let v p be the normaized exponentia vauation of C p with p p = p vp(p = 1 p (see[1-9]. When one taks of q-extension, q is variousy considered as an indeterminate, a compex q C or a p-adic number q C p.ifq C, then one normay assumes q < 1. If q C p, then we assume that p < 1. Aso this paper, we use the foowing notation: [x] q = x, [x] q = 1 ( qx. (1.1 1+q Hence im q 1 [x] q = x for a x Z p. For g UD(Z p, Kim defined the q-deformed fermionic p-adic integra on Z p 1 I q (g = g(xdμ q (x = im g(x( q x. Z N p [p N ] (1.2 q 0<a<dp Let g 1 (x be the transation with g 1 (x =g(x+1. Then we have the foowing integra equation: n 1 q n I q (g n +( 1 n 1 I q (g = [2] q ( 1 n 1 q (. (1.3 For g UD(Z p, g(xdμ q (x = g(xdμ q (x. (1.4 Z p We introduced generaized Genocchi number and poynomias. Let χ be a primitive Dirichet character of conductor f N. We assume that f is odd. Then the generaized Genocchi numbers associated with χ are defined by F χ (t = 2t χ(i( 1i e it t n = G e ft n,χ +1. (1.5 by The generaized Genocchi poynomias associated with χ are aso defined F χ (t, x = 2t χ(i( 1i e it e tx = e ft +1 G n,χ (x tn. (1.6

3 Generaized q-genocchi numbers and poynomias 5853 In the specia case x = 0, G n,χ = G n,χ (0 are caed the n-th generaized Genocchi numbers attached to χ. These numbers and poynomias are interpoated by the q-genocchi zeta function and Hurwitz q-genocchi zeta function, respectivey. 2 Generaized q-genocchi numbers and poynomais with weak weight α Our primary goa of this section is to define the generaized q-genocchi numbers and poynomias with weak weight α. We aso find generating functions of the generaized q-genocchi numbers and poynomias with weak weight α. These poynomias wi be used to prove the anaytic continuation of the generaized Hurwitz q-genocchi zeta function. First, we introduce the generaized q-genocchi numbers with weak weight α. Definition 2.1 For q C p with p < 1, α Z, n,χ,q = χ(xn[x] q n 1 dμ q α(x. (2.1 By using p-adic q-integra, we have χ(xn[x] n 1 q dμ q α(x = [2] q αn ( 1 i q iα χ(i By (2.1, we obtain n,χ,q =[2] q αn ( 1 i q iα χ(i n 1 n 1 n 1 n 1 ( 1 q i 1 1+q (α+f. In order to find the generating function of n,χ,q, we set By using (2.3, we have F (α n,χ,q = [2] q αn ( 1 i q iα χ(i = [2] q αn χ,q (t = ( 1 q α χ([] n 1 q. t n n,χ,q n 1 n 1 (2.2 ( 1 q i 1 1+q (α+f. (2.3. (2.4 ( 1 q i 1 1+q (α+f (2.5

4 5854 J. Y. Kang and C. S. Ryoo By using (2.4 and (2.5, we obtain that ( F χ,q (α (t = [2] q αn ( 1 q α χ([] q n 1 = [2] q αt ( 1 q α χ(e []qt. t n (2.6 Then generaized q-genocchi numbers G n,χ,q (α with weak weight α are defined by means of the generating function F χ,q (α (t = [2] q αt ( 1 n q αn χ(ne [n]qt = G n,χ,q (α t n. (2.7 Remark 2.2 In (2.7, we see that im F χ,q (α (t =2t ( 1 n χ(ne nt = 2t χ(i( 1i e it q 1 e ft +1 By using (2.1, we obtain t n n,χ,q = χ(xn[x] q n 1 dμ q α(x tn = tχ(xe [x]qt dμ q α(x. = F χ (t. (2.8 (2.9 By (2.7 and (2.9, we have tχ(xe [x]qt dμ q α(x = [2] q αt ( 1 n q αn χ(ne [n]qt. (2.10 Next, we introduce the generaized q-genocchi poynomias with weak weight α. Definition 2.3 For q C p with p < 1, α Z, n,χ,q (x = χ(yn[x + y] n 1 q dμ q α(y. (2.11 By using p-adic q-integra, we have χ(yn[x + y] n 1 q dμ q α(y = [2] q αn ( 1 i q iα χ(i n 1 n 1 ( 1 q i q x 1 1+q (α+f. (2.12

5 Generaized q-genocchi numbers and poynomias 5855 By using (2.11 and (2.12, we get n,χ,q(x = [2] q αn ( 1 i q iα χ(i n 1 n 1 In order to find the generating function of n,χ,q(x, we set By using (2.13, we obtain F (α χ,q (t, x = ( 1 q i q x 1 1+q (α+f. (2.13 n,χ,q (xtn. (2.14 n,χ,q(x = [2] q αn ( 1 i q iα χ(i = [2] q αn n 1 n 1 ( 1 q α χ([x + ] n 1 q. By using (2.14, we get ( F χ,q (α (t, x = [2] q αn ( 1 q α χ([x + ] q n 1 = [2] q αt ( 1 q α χ(e [x+]qt. Hence, we are defined the generating function of n,χ,q(x F (α χ,q (t, x = [2] q αt ( 1 n q αn χ(ne [x+n]qt = ( 1 q i q x 1 1+q (α+f. t n (2.15 (2.16 n,χ,q (xtn. (2.17 Remark 2.4 In (2.17, we observe that im F χ,q (α (t, x =2t q 1 ( 1 n χ(ne (x+nt = 2t χ(i( 1i e it e xt = F e ft χ (t, x. +1 (2.18

6 5856 J. Y. Kang and C. S. Ryoo 3 Some reations reated to n,χ,q and n,χ,q(x In this section, we investigate some reations reated to G n,χ,q (α and G n,χ,q(x. (α Since [x + y] q =[x] q + q x [y] q, we have n+1,χ,q(x =(n +1 χ(y[x + y] n q dμ q α(y = q ( (3.1 x [x] q + q x G χ,q (α n+1. Aso, we get ( n,χ,q(x tn = n 1 ( n 1 χ(yn ( n 1 ( n 1 = n [x] n 1 q q By comparing coefficient tn Theorem 3.1 Let n N. Then we have n,χ,q (x = [x] q n 1 q x [y] qdμ q α(y G(α x +1,χ,q +1 in (3.2, we have the foowing theorem. n k=0 t n t n. (3.2 ( n [x] n k q q x(k 1 k,χ,q k. (3.3 By using (2.1 and (2.11, we have the foowing theorem. Theorem 3.2 Let n N. We have n,χ,q = [f]n 1 q [f] q α n,χ,q(x = [f]n 1 q [f] q α By using (1.7, we easiy see that χ(i( 1 i q iα n,q f ( i f, χ(i( 1 i q iα n,q f ( i + x f. n q αnf G α m,χ,q (nf+( 1n G m,χ,q (α = [2] q αm ( 1 n q α χ([] m 1 Hence, we have the foowing theorem. q.

7 Generaized q-genocchi numbers and poynomias 5857 Theorem 3.3 Let m Z +.Ifn 0 (mod 2, then n q αnf m,χ,q(nf m,χ,q = [2] q αm ( 1 +1 q α χ([] m 1 q, If n 1 (mod 2, then n q αnf m,χ,q(nf+g m,χ,q (α = [2] q αm ( 1 q α χ([] m 1 q. 4 The q- Genocchi zeta function In the section,we assume that q C with q < 1. By using generaized q- Genocchi numbers and poynomias with weak weight α, generaized q-genocchi zeta function and generaized Hurwitz q-genocchi zeta function are defined. These functions interpoate the generaized q-genocchi numbers with weak weight α and the generaized q-genocchi poynomias with weak weight α, respectivey. From (2.7, we note that d k+1 (α F dtk+1 χ,q t=0 (t = [2] q α(k +1 ( 1 n q αn χ(n[n] k q, (k N. (4.1 Definition 4.1 For s C, we define ζ (α χ,q (s = [2] q α n=1 Note that ζ (α χ,q (s is a meromorphic function on C. ( 1 n q αn χ(n. (4.2 [n] s q Remark 4.2 Let s C. Then we have Reation between ζ (α χ,q (s and k,χ,q Theorem 4.3 For k N, we have im ζ χ,q (α (s =2 ( 1 n χ(n. (4.3 q 1 n s n=1 is given by the foowing theorem. ζ (α χ,q ( k = 1 k +1 G(α k+1,χ,q. (4.4

8 5858 J. Y. Kang and C. S. Ryoo Observe that ζ χ,q (α (s interpoates k,χ,q at non-negative integers. By using (2.17, we note that d k+1 (α F dtk+1 χ,q (t, x = [2] q α(k +1 ( 1 q α χ([x + ] k q, (k N. (4.5 t=0 By (4.5, we are now ready to define the generaized Hurwitz q- Genocchi zeta functions. Definition 4.4 Let s C. Then we have ζ χ,q (α (s, x = [2] ( 1 q α χ( q α. [x + ] s (4.6 q Note that ζ χ,q (α (s, x is a meromorphic function on C. Reation between ζ χ,q (α (s, x and k,χ,q (x is given by the foowing theorem. Theorem 4.5 For k N, we get ζ χ,q (α 1 ( k, x = k +1 G(α k+1,χ,q (x. (4.7 Observe that ζ χ,q (α ( k, x function interpoates k+1,χ,q (x at non-negative integers. ACKNOWLEDGEMENTS. This paper has been supported by the 2012 Hannam University Research Fund. Correspondence shoud be addressed to C. S. RYOO, ryoocs@hnu.kr. References [1] M. Cenkci, M. Can, V. Kurt, q-adic interpoation functions and kummer type congruence for q-twisted and q-generaized twisted Euer numbers, Advan. Stud. Contemp. Math., 9(2004, [2] L. C. Jang, A study on the distribution of q-genocchi poynomias, Adv. Stud. Contemp. Math., 18(2009, [3] L. C. Jang, T. Kim and H. Pak, A note on q-euer and Genocchi numbers, Proc. of Japan Academy, Ser A 77(2001, [4] T. Kim, L. C. Jang, and H. K. Pak, A note on the q-genocchi numbers and poynomias, Journa of Inequaities and Appications, 2007(2007, 8 pages, Artice ID

9 Generaized q-genocchi numbers and poynomias 5859 [5] T. Kim, J. Choi, Y-.H. Kim, C. S. Ryoo, A Note on the weighted p-adic q-euer measure on Z p, Advan. Stud. Contemp. Math., 21(2011, [6] C. S. Ryoo, A numerica computation on the structure of the roots of q-extension of Genocchi poynomias, Appied Mathematics Letters, 21(2008, [7] C. S. Ryoo, On the generaized Barnes type mutipe q-euer poynomias twisted by ramified roots of unity, Proc. Jangjeon Math. Soc., 13(2010, [8] C. S. Ryoo, A note on the weighted q-euer numbers and poynomias, Advan. Stud. Contemp. Math., 21(2011, [9] Y. Simesk, V. Kurt, D. Kim, New approach to the compete sum of products of the twisted (h, q-bernoui numbers and poynomias, J. Noninear Math. Phys., 14(2007, Received: June, 2012