Serkan Araci, Mehmet Acikgoz, and Aynur Gürsul

Transcription

1 Commun. Korean Math. Soc ), No. 3, pp ANALYTIC CONTINUATION OF WEIGHTED q-genocchi NUMBERS AND POLYNOMIALS Serkan Araci, Mehmet Acikgoz, and Aynur Gürsul Abstract. In the present paper, we analyse analytic continuation of weighted q-genocchi numbers and polynomials. A novel formula for weighted q-genocchi-zeta function ζ G,q s α) in terms of nested series of ζ G,q n α) is derived. Moreover, we introduce a novel concept of dynamics of the zeros of analytically continued weighted q-genocchi polynomials. 1. Introduction In this paper, we use notations like N, R and C, where N denotes the set of natural numbers, R denotes the field of real numbers and C also denotes the set of complex numbers. When one talks of q-extension, q is variously considered as an indeterminate, a complex number or a p-adic number. Throughoutthis work, we will assumethat q C with q < 1. The q-integer symbol [x : q] denotes as [x : q] qx 1 q 1 see [1-10]). Firstly, analytic continuation of q-euler numbers and polynomials was investigated by Kim in [8]. He gavea new concept of dynamics of the zeros of analytically continued q-euler polynomials. Actually, we are motivated from his excellent paper which is Analytic continuation of q-euler numbers and polynomials, Applied Mathematics Letters ), By the same motivation, we also procure the analytic continuation of weighted q-genocchi numbers and polynomials as parallel to his article. However, we give some interesting identities by using generating function of weighted q-genocchi polynomials. Received October 11, Mathematics Subject Classification. Primary 05A10, 11B65; Secondary 11B68, 11B73. Key words and phrases. Genocchi numbers and polynomials, q-genocchi numbers and polynomials, weighted q-genocchi numbers and polynomials, weighted q-genocchi-zeta function. 457 c 2013 The Korean Mathematical Society

2 458 SERKAN ARACI, MEHMET ACIKGOZ, AND AYNUR GÜRSUL 2. Some properties of the weighted q-genocchi numbers and polynomials For α N {0}, the weighted q-genocchi polynomials are defined by means of the following generating function: For x C, 2.1) G n,q x α) tn n! [2 : q]t 1) n q n e t[n+x:qα]. In the special case, x 0 in 2.1), Gn,q 0 α) : G n,q α) are called the weighted q-genocchi numbers. By 2.1), we readily derive the following: G,q x α) [2 : q] n n 2.2) ) 1) [α : q] n 1 q) n l q αlx l 1+q αl+1, where n l) is the binomial coefficient. By expression 2.1), we see that 2.3) Gn,q x α) q αx q αx Gq α)+[x : q α ]) n, with the usual convention of replacing l0 ) n Gq α) by Gn,q α) is used for details, see [1], [2]). α) Let T q x, t) be the generating function of weighted q-genocchi polynomials as follows: 2.4) Tα) q x,t) G n,q x α) tn n!. Then, we easily notice that 2.5) Tα) q x,t) [2 : q]t 1) n q n e t[n+x:qα]. From expressions 2.4) and 2.5), we procure the followings: For k even) and n,α N {0}, we have 2.6) q k G,q k α) G,q α) For k odd) and n,α N {0}, we have 2.7) q k G,q k α) + G,q α) j0 k 1 [2 : q] 1) l+1 q l [l : q α ] n. l0 k 1 [2 : q] 1) l q l [l : q α ] n. Via Eq.2.5), we easily obtain the following: n ) n 2.8) Gn,q x α) q αx q αjx Gj,q α)[x : q α ] n j. j l0

3 ANALYTIC CONTINUATION 459 From 2.6)-2.8), we get the following: 2.9) k 1 [2 : q] 1) l+1 q l [l : q α ] n l0 q k1+αn) 1 ) G,q α) + q1 α)k n ) q αjk Gj,q α)[k : q α ] j, j where k is an even positive integer. If k is an odd positive integer. Then, we can derive the following equality: 2.10) k 1 [2 : q] 1) l q l [l : q α ] n l0 q k1+αn) +1 ) G,q α) + q1 α)k j0 n ) q αjk Gj,q α)[k : q α ] j. j j0 3. On the weighted q-genocchi-zeta function The famous Genocchi polynomials are defined by 2t 3.1) e t +1 ext G n x) tn, t < πcf. [6]). n! For s C, x R with 0 x < 1, Genocchi-Zeta function is given by 1) n 3.2) ζ G s,x) 2 n+x) s, and 3.3) ζ G s) n1 1) n n s. By3.1), 3.2) and3.3), Genocchi-zeta functions are related to the Genocchi numbers as follows: ζ G n) G. Moreover, it is simple to see ζ G n,x) G x). The weighted q-genocchi Hurwitz-zeta type function is defined by 1) m q m ζg,q s,x α) [2 : q] [m+x : q α ] s. m0

4 460 SERKAN ARACI, MEHMET ACIKGOZ, AND AYNUR GÜRSUL Similarly, weighted q-genocchi-zeta function is given by 1) m q m ζg,q s α) [2 : q] [m : q α ] s. For n,α N {0}, we have m1 ζg,q n α) G,q α). We now consider the function G q n : α) as the analytic continuation of weighted q-genocchi numbers. All the weighted q-genocchi numbers agree with G q n : α), the analyticcontinuationofweightedq-genocchinumbersevaluated at n. For n 0, G q n : α) G n,q α). We cannowstate G qs : α) in termsof ζ G,q s α), the derivativeof ζ G,q s : α) G q s+1 : α) s+1 ζ G,q s α), For n,α N {0} G qs+1 : α) s+1 ζ G,q s α). G q2 : α) 2 ζ Ǵ,q 2n α). This is suitable for the differential of the functional equation and so supports the coherence of G q s : α) and G qs : α) with G n,q α) and ζ G,q s α). From the analytic continuation of weighted q-genocchi numbers, we derive as follows: G q s+1 : α) s+1 ζ G,q s α) and G q s+1 : α) s+1 Moreover, we derive the following: For n N {1} G,q α) G q : α) ζ G,q n α). ζ G,q s α). The curve G q s : a) review quickly the points G s,q α) and grows n asymptotically n). The curve G q s : a) review quickly the point G q s : a). Then, we procure the following: lim G q : α) lim ζg,q n α) lim [2 : q] m1 lim q 2[ 2 : q 1]. ) 1) m q m [m : q α ] n q[2 : q]+[2 : q] m2 ) 1) m q m [m : q α ] n

5 ANALYTIC CONTINUATION 461 From this, we easily note that G q : α) ζ G,q n α) G q s+1 : α) s+1 ζ G,q s α). 4. Analytic continuation of the weighted q-genocchi polynomials For coherence with the redefinition of G n,q α) G q n : α), we have G n ) n n,q x α) q αx q αkx Gk,q α)[x : q α ] n k. k k0 Let Γs) be Euler-gamma function. Then the analytic continuation can be get as n k) Γ) Γn k+1)γk+1) n s R, x w C, G n,q α) G q k +s [s] : α) ζ G,q k +s [s]) α), G s,q w α) G q s,w : α) Γs+1) Γ1+k +s [s]))γ1+[s] k) [s] q αw k 1 [s]+1 q αw k0 Γs+1) G q k +s [s]) : α)q αwk+s [s])) Γ1+k +s [s]))γ1+[s] k) [w : q α ] [s] k Γs+1) G q 1+k +s [s]) : α)q αwk 1+s [s])) [w : q α ] [s]+1 k. Γk +s [s]))γ2+[s] k) Here [s] gives the integer part of s, and so s [s] gives the fractional part. Deformation of the curve G q 1,w : α) into the curve of G q 2,w : α) is by means of the real analytic cotinuation G q s,w : α), 1 s 2, 0.5 w 0.5. References [1] S. Araci, M. Acikgoz, and J. J. Seo, A study on the weighted q-genocchi numbers and polynomials with their interpolation function, Honam Math. J ), no. 1, [2] S. Araci, D. Erdal, and J. J. Seo, A study on the fermionic p-adic q-integral representation on Z p associated with weighted q-bernstein and q-genocchi polynomials, Abstr. Appl. Anal ), Article ID , 10 pages. [3] T. Kim, On explicit formulas of p-adic q-l-functions, Kyushu J. Math ), no. 1, [4], On a q-analogue of the p-adic log gamma functions and related integrals, J. Number Theory ), no. 2, [5], Power series and asymptotic series associated with the q-analog of the twovariable p-adic L-function, Russ. J. Math. Phys ), no. 2, [6], On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl ), no. 2, [7], On the analogs of Euler numbers and polynomials associated with p-adic q- integral on Z p at q 1, J. Math. Anal. Appl ), no. 2, [8], Analytic continuation of q-euler numbers and polynomials, Appl. Math. Lett ), no. 12,

6 462 SERKAN ARACI, MEHMET ACIKGOZ, AND AYNUR GÜRSUL [9], On p-adic interpolating function for q-euler numbers and its derivatives, J. Math. Anal. Appl ), no. 1, [10] T. Kim, C. S. Ryoo, L. C. Jang, and S. H. Rim, Exploring the q-riemann zeta function and q-bernoulli polynomials, Discrete Dyn. Nat. Soc ), no. 2, Serkan Araci Department of Mathematics Faculty of Arts and Science University of Gaziantep Gaziantep, Turkey address: Mehmet Acikgoz Department of Mathematics Faculty of Arts and Science University of Gaziantep Gaziantep, Turkey address: Aynur Gürsul Department of Mathematics Faculty of Arts and Science University of Gaziantep Gaziantep, Turkey address: