On Symmetric Property for q-genocchi Polynomials and Zeta Function

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1 Int Journal of Math Analysis, Vol 8, 2014, no 1, 9-16 HIKARI Ltd, wwwm-hiaricom On Symmetric Property for -Genocchi Polynomials and Zeta Function J Y Kang Department of Mathematics Hannam University, Daejeon , Korea C S Ryoo Department of Mathematics Hannam University, Daejeon , Korea Copyright c 2014 J Y Kang and 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 wor is properly cited Abstract The present papers deal with the various -Genocchi numbers and polynomials We find the symmetric identity of -Genocchi zeta function By using the symmetric identity of -Genocchi zeta function, we study a few interesting symmetric properties of -Genocchi polynomials Mathematics Subject Classification: 11B68, 11S40, 11S80 Keywords: symmetric identities of -Genocchi polynomials, symmetric property of -Genocchi zeta function 1 Introduction The Genocchi numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics The classical Genocchi polynomials are defined by F (t, x = 2t e t +1 ext = e G(xt = G n (x tn, t <π, n!

2 10 J Y Kang and C S Ryoo with the usual convention of replacing G n (x byg n (x G n (0 = G n are called the n-th Genocchi numbers For s C and Re(s > 0, the Hurwitz-type Genocchi zeta function are defined by ζ G (s, x =2 ( 1 n, (see [5], (n + x s Note that the Hurwitz-type Genocchi zeta function has the values of the Genocchi polynomials at negative integers Recently, many mathematicians have studied in the area of the -Genocchi numbers and polynomials(see [1-16] When one tals of -extension, is considered in many ways such as an indeterminate, a complex number C, or p-adic number C p Let us assume that C with < 1 Then we use the notation: [x] = 1 x 1, [x] = 1 ( x 1+ (cf [1-5] Note that lim 1 [x] = x for any x In this paper, we consider that C notes the complex plane Definition 11 (generating function of the -Genocchi polynomials Let C Then we define G n, (x tn n! = [2] t ( 1 m m e [x+m]t, where we use the technical method s notation by replacing G n (x by G n (x, symbolically (see [1-5] In the special case x =0, G n, (0 = G n, are called the n-th -Genocchi numbers(see [1-5] Definition 12 (the Hurwitz-type -Genocchi zeta function For s C and Re(s > 0, ( 1 n n ζ (s, x = [2], (see [5], [n + x] s Mathematicians also found out various properties of the -Genocchi polynomials as follows:(see [1-5] Theorem 13 (the -Genocchi polynomials Let C Then we have G n, (x = [2] n ( 1 m m [x + m] n 1

3 Symmetric property for -Genocchi polynomials and zeta function 11 Theorem 14 (the addition theorem for the -Genocchi polynomials Let n be a positive integer Then one has ( n G n, (x + y = lx G l, (y[x] n l l l=0 Theorem 15 (symmetric distribution for the -Genocchi polynomials Let n 0 Then we get G n, 1(1 x =( n 1 G n, (x Theorem 16 (the relation of -Genocchi polynomials and zeta function For N, we have ζ (, x = G +1,(x +1 Observe that ζ (, x function interpolates G, (x polynomials at non-negative integers Our aim in this paper is to discover special symmetric properties for - Genocchi polynomials We are going to find a symmetric identity for - Genocchi zeta function From property of the -Genocchi zeta function, we derive some symmetric properties of -Genocchi polynomials by combing the basic Theorem and some formulas 2 Symmetry properties of Genocchi numbers and polynomials with wea weight α In this section, one of the most important theorems is the Theorem 21 It will be used to obtain the main results of -Genocchi polynomials We also establish several interesting symmetric identities for -Genocchi polynomials and -Genocchi zeta function Theorem 21 Let s C with Re(s > 0 and a,b : odd positive integers Then one has [2] a[a] s ( 1 i ai ζ b(s, ax + ai b = [2] b[b]s ( 1 j bj ζ a(s, bx + bj a i=0 Proof By substitute ax + ai for x in Definition 12 and replace by b,we b derive ζ b(s, ax + ai b = [2] ( 1 n bn b [ax + ai + = [2] b[b] s ( 1 n bn b n]s [abx + ai + bn] s b j=0

4 12 J Y Kang and C S Ryoo Since for any non-negative integer m and odd positive integer a, there exist uniue non-negative integer r such that m = ar +j with 0 j a 1 Hence, this can be written as ζ b(s, ax + ai b = [2] b[b]s = [2] b[b] s ar+j=0 0 j a 1 It follows from the above euation that j=0 r=0 ( 1 ar+j b(ar+j [b(ar + j+abx + ai] s ( 1 ar+j abr+bj [ab(r + x+ai + bj] s [2] a[a] s ( 1 i ai ζ b(s, ax + ai b i=0 = [2] a[2] b[a] s [b] s i=0 j=0 r=0 ( 1 ar+i+j abr+ai+bj [ab(r + x+ai + bj] s (21 In the similar method, we can have that ζ a(s, bx + bj a = [2] a ( 1 n an [bx + bj a + n]s a = [2] a[a] s ( 1 n an, [abx + bj + an] s [2] b[b] s ( 1 j bj ζ a(s, bx + bj a j=0 = [2] a[2] b[a] s [b]s i=0 j=0 r=0 ( 1 br+i+j abr+ai+bj [ab(r + x+ai + bj] s (22 Thus, we complete the proof of the theorem by combining (21 and (22 In Theorem 21, we get the following formulas for the -Genocchi zeta function Corollary 22 Let b =1in Theorem 21 Then we have ( 1 j j ζ a(s, x + j a =[a] 2[a]s 1 ζ (s, ax j=0 Note that if 1, then a 1 j=0 ( 1j j ζ a(s, x + j a =as ζ(s, ax Corollary 23 Let a =2,b=1in Theorem 21 Then we get ζ 2(s, x ζ 2(s, x = (1 + [2]s ζ (s, 2x

5 Symmetric property for -Genocchi polynomials and zeta function 13 We can easily see that if 1, then ζ(s, x =ζ(s, x s ζ(s, 2x By Theorem 16 and Theorem 21, we have the following theorem Theorem 24 Let a, b be any odd positive integer and s, t be non-negative integer Then for non-negative integers n, one has [2] a[b] n 1 s=0 ( 1 s as G n, b(ax + as b = [2] b[a]n 1 Considering a = 1 in the Theorem 24, we obtain as below t=0 ( 1 t bt G n, a(bx + bt a ( 1 s s G n, b(x + s b = s=1 1+ b (1 + [b] n 1 G n, (bx From now on, we obtain another result by applying the addition theorem for the -Genocchi polynomials(theorem 14 Theorem 25 Let a, b be any odd positive integer and s, t be non-negative integer ( n [2] a [a] [b] n 1 G n n, b(ax ( 1 s (n+1 as [s] a s=0 ( n = [2] b [a] n 1 [b] G n, a(bx ( 1 t (n+1 bt [t] n b 3 Some symmetric properties of -Genocchi polynomials In this section, we derive the symmetric results by using definition and theorem of -Genocchi polynomials The results are able to express very well the symmetric property of -Genocchi polynomials By using Definition 11 and after some elementary calculations, we have the following theorem Theorem 31 Let n, m be non-negative integer Then we obtain that ( n m = t=0 ( + m (n x [ x] n G +m 1, (x + y ( m ( + n (n+ 1x [x] m G +n 1, (y

6 14 J Y Kang and C S Ryoo Proof By using Definition 11, we easily see that [2] (u+v ( 1 m m e [x+y+m](u+v = [2] (u+ve [x](u+v Since [x + y] =[x] + x [y], we have ( 1 m m e x [y+m] (u+v e [x]v [2] (u + v ( 1 m m e [x+y+m](u+v = e [x]u [2] (u + v ( 1 m m e x [y+m] (u+v (31 The left-hand side of (31 can be expressed as e [x]v [2] (u + v = ( ( 1 m m e [x+y+m](u+v ( n ( + m (n x G +m 1, (x + y[ x] n u m v n m! n! (32 The right-hand side of (31 can be expressed as follows: e [x]u [2] (u + v = ( m ( 1 m m e x [y+m] (u+v ( m ( + n (+n 1x G +n 1, (y[x] m v n u m n! m! (33 By comparing the coefficients of vn u m in (32 and (33, we assert that the n!m! theorem is right The Theorem 31 that was made by the addition Theorem 14 and the Definition 11 is very useful to find the symmetric identity of -Genocchi polynomials Theorem 32 Let s, t be non-negative integer Then we have l m ( ( l m s t s=0 t=0 ( n = s s=0 ( [x + y] l+m s t (n + s + tg n+s+t 1, (x + y + z (l+m+s 1xy [x + y] n s (l + m + sg l+m+s 1, (z

7 Symmetric property for -Genocchi polynomials and zeta function 15 By applying the symmetric distribution for the -Genocchi polynomials(theorem 15 in Theorem 31, we also get the following theorem Theorem 33 Let, n, m be non-negative integer Then we have m+1 ( 1 m =( 1 n+1 n+1 ( m +1 ( n +1 ( + n(1 + + n +n 1 G +n 1, (x + m(1 + + m (+m 2 G +m 1, 1( x Proof By using the Theorem 15, we get the following euation m ( m ( 1 m ( + n +n 1 G +n 1, (x ( n =( 1 n ( + m (+m 2 G +m 1, 1( x (34 By using (34, we get the following euations ( 1 m (n +1 m+1 n+1 =( 1 n+1 ( m +1 ( n +1 m+1 ( m +1 ( 1 m (m +1 1 n+1 ( n +1 =( 1 n+1 (m +1 ( + n +n 1 G +n 1, (x ( + m (+m 2 G +m 1, 1( x, ( + n +n 1 G +n 1, (x ( + m (+m 2 G +m 1, 1( x (35 (36 Thus, we conclude the following result by applying (35, (36 in (34 ACKNOWLEDGEMENTS This wor was supported by NRF(National Research Foundation of Korea Grant funded by the Korean Government(NRF Fostering Core Leaders of the Future Basic Science Program References [1] Ismail Naci Cangul, Hacer Ozdena, Yilmaz Simse, A new approach to - Genocchi numbers and their interpolation functions, Nonlinear Analysis: Theory, Methods and Applications, 71( 2009,

8 16 J Y Kang and C S Ryoo [2] Min-Soo Kim, Su Hu, On p-adic Hurwitz-type Euler zeta function, J Number Theory, 132(2012, [3] T Kim, Symmetry p-adic invariant integral on Z p for Bernoulli and Euler polynomials, J Difference Eu Appl, 14(2008, [4] C S Ryoo, A numerical computation on the structure of the roots of (h, -extension of Genocchi polynomials, Mathematical and Computer Modelling, 49(2009, [5] C S Ryoo, J Y Kang, On the twisted -Genocchi numbers and polynomials with wea weight α, Applied Mathematical Science (6( , Received: November 1, 2013