A Note on Carlitz s Twisted (h,q)-euler Polynomials under Symmetric Group of Degree Five

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1 Gen. Math. Note, Vol. 33, No. 1, March 2016, pp.9-16 ISSN ; Copyright c ICSRS Publication, Available free online at A Note on Carlitz Twited (h,)-euler Polynomial under Symmetric Group of Degree Five Ugur Duran 1 and Mehmet Acikgoz 2 1,2 Univerity of Gaziantep, Faculty of Science and Art Department of Mathematic, TR Gaziantep, Turkey 1 duran.ugur@yahoo.com 2 acikgoz@gantep.edu.tr (Received: / Accepted: ) Abtract In 12], Ryoo defined the Carlitz twited (h, )-Euler number and polynomial. In thi paper, we conider ome new ymmetric identitie for Carlitz twited (h, )-Euler polynomial ariing from the fermionic p-adic integral over the p-adic number field under the ymmetric group of degree five. Keyword: Carlitz twited (h, )-Euler polynomial, Invariant under S, p-adic invariant integral on, Symmetric identitie. 1 Introduction In the Taylor expanion n=0 E n (x) tn n! = 2 e t + 1 ext with ( t < π), (1) E n (x) denote the n-th Euler polynomial. If we take x = 0 in the E. (1), we then have E n (0) := E n that i commonly known a the n-th Euler number (ee, e.g., 6, 8, 9, 11-13]). Let N = {1, 2, 3, }, N = N {0} and p be choen a a fixed odd prime number. Along thi paper, Q, Q p and C p hall denote topological cloure of Z, the field of rational number, topological cloure of Q and the field of p-adic completion of an algebraic cloure of Q p, repectively.

2 10 Ugur Duran et al. For d an odd poitive number with (d, p) = 1, let X := X d = lim N Z/dp N Z and X 1 = and t + dp N = { x X x t ( moddp N)} in which t Z lie in 0 t < dp N. See, for more information, 1-13]. The normalized abolute value according to the theory of p-adic analyi i given by p p = p 1. The notation can be conidered a an indeterminate, a complex number C with < 1, or a p-adic number C p with 1 p < p 1 p 1 and x = exp (x log ) for x p 1. For fixed x, let u introduce the following notation (ee 1-13]): x] = 1 x 1 (2) which i known a -number of x (or -analogue of x). We note that a 1, the notation x] reduce to the x. For f UD ( ) = {f f : C p i uniformly differentiable function}, Kim 8] defined the p-adic invariant integral on a follow: I 1 (f) = From E. (3), we get f (x) dµ 1 (x) = lim N p N 1 x=0 n 1 I 1 (f n ) = ( 1) n I 1 (f) + 2 ( 1) n k 1 f(k) f (x) ( 1) x. (3) where f n (x) implie f(x + n). For more detail about thi topic, take a look at the reference 2, 3, 8, 12, 13]. Let h Z and where C p N = { w : w pn = 1 T p = N 1 C p N = lim C p N N, } i the cyclic group of order p N. For w T p, we how by φ w : C p the locally contant function x w x. For C p with 1 p < 1 and w T p, the Carlitz twited (h, )-Euler polynomial are defined by the following p-adic fermionic integral on in 12]: E n,,w(x) (h) = w y hy x + y] n dµ 1 (y) (n 0). (4)

3 A Note on Carlitz Twited (h,)-euler Polynomial If we let x = 0 into the E. (4), we get E n,,w(0) (h) := E n,,w (h) called n-th Carlitz twited (h, )-Euler number. Taking w = 1 and 1 in the E. (4) yield to E n,,w(x) (h) E n (x) := (x + y) n dµ 1 (y). Recently, ymmetric identitie of ome pecial polynomial, uch a - Genocchi polynomial of higher order under third Dihedral group D 3 in 1], -Genocchi polynomial under the ymmetric group of degree four in 4], weighted -Genocchi polynomial under the ymmetric group of degree four in ], -Frobeniou-Euler polynomial under ymmetric group of degree five in 3], Carlitz -type -Euler polynomial invariant under the ymmetric group of degree five in 9], higher-order Carlitz -Bernoulli polynomial under the ymmetric group of degree five in 10], have been tudied by many mathematician. In the following ection, we invetigate ome new ymmetric identitie for Carlitz twited (h, )-Euler polynomial ariing from the p-adic invariant integral on under the ymmetric group of degree five denoted by S. 2 Symmetric Identitie for E (h) n,,w(x) under S Let h Z, w i N be a natural number which atifie the condition w i 1(mod2), in which i Z lie in 1 i. By the E. (3) and (4), we acuire w w 1w 2 w 3 w 4 y hw 1w 2 w 3 w 4 y () e w 1w 2 w 3 w 4 y+ w x+w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 ] t dµ 1 (y) p N 1 = lim ( 1) y w w 1w 2 w 3 w 4 y hw 1w 2 w 3 w 4 y N y=0 e w 1w 2 w 3 w 4 y+ w x+w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 ] t = lim N 1 l=0 ( 1) l+y w w 1w 2 w 3 w 4 (l+w y) hw 1w 2 w 3 w 4 (l+w y) p N 1 y=0 e w 1w 2 w 3 w 4 (l+w y)+ w x+w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 ] t. Taking ( 1) i+j+k+ w w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 =0 h(w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 )

4 12 Ugur Duran et al. on the both ide of E. () give ( 1) i+j+k+ w w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 =0 h(w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 ) w w 1w 2 w 3 w 4 y hw 1w 2 w 3 w 4 y e w 1w 2 w 3 w 4 y+ w x+w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 ] t dµ 1 (y) = lim N =0 1 l=0 p N 1 ( 1) i+j+k++y+l y=0 w w 1w 2 w 3 w 4 (l+w y)+w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 h(w 1w 2 w 3 w 4 (l+w y)+w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 ) e w 1w 2 w 3 w 4 (l+w y)+ w x+w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 ] t. (6) Oberve that the euation (6) i invariant for any permutation σ S. Therefore, we obtain the following theorem. Let h Z, w i N be a natural number which atifie the condition w i 1(mod2), in which i Z lie in 1 i and n 0. Then the following w σ(1) 1 w σ(2) 1 w σ(3) 1 w σ(4) 1 =0 ( 1) i+j+k+ w w σ()w σ(4) w σ(2) w σ(3) i+w σ() w σ(4) w σ(1) w σ(3) j+w σ() w σ(4) w σ(1) w σ(2) k+w σ() w σ(3) w σ(1) w σ(2) ( ) h wσ() w σ(4) w σ(2) w σ(3) i+w σ() w σ(4) w σ(1) w σ(3) j+w σ() w σ(4) w σ(1) w σ(2) k+w σ() w σ(3) w σ(1) w σ(2) w w σ(1)w σ(2)w σ(3)w σ(4)(l+w σ()y) hw σ(1)w σ(2)w σ(3)w σ(4)(l+w σ()y) exp ( w σ(1) w σ(2) w σ(3) w σ(4) y + w σ(1) w σ(2) w σ(3) w σ(4) w σ() x +w σ() w σ(4) w σ(2) w σ(3) i + w σ() w σ(4) w σ(1) w σ(3) j +w σ() w σ(4) w σ(1) w σ(2) k + w σ() w σ(3) w σ(1) w σ(2) ] t ) dµ 1 (y) hold true for any σ S. By uing E. (2), we have y + w x + w w 4 w 2 w 3 i (7) +w w 4 w 1 w 3 w 4 w 1 w 2 k + w w 3 w 1 w 2 ] = ] y + w x + w k + w ].

5 A Note on Carlitz Twited (h,)-euler Polynomial From E. () and (7), we derive e w 1w 2 w 3 w 4 y+ w x+w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 ] t w w 1w 2 w 3 w 4 y hw 1w 2 w 3 w 4 y dµ 1 (y) = ] n n=0 w w 1w 2 w 3 w 4 y hw 1w 2 w 3 w 4 y Z p y + w x + w k + w ] n = ] n n=0 E (h) n,,w w 1 w 2 w 3 ww 4 dµ 1 (y) tn n! ( w x + w k + w ) t n n!. (8) By E. (8), for n 0,we have y + w x + w w 4 w 2 w 3 i (9) +w w 4 w 1 w 3 w 4 w 1 w 2 k + w w 3 w 1 w 2 ] n w w 1w 2 w 3 w 4 y hw 1w 2 w 3 w 4 y dµ 1 (y) = ] n E (h) n,,w ( w x + w k + w ). Thu, from Theorem 2 and E. (9), we have the following theorem. Let h Z, w i N be a natural number which atifie the condition w i 1(mod2), in which i Z lie in 1 i and n 0. Hence, the following wσ(1) w σ(2) w σ(3) w σ(4) ] n w σ(1) 1 w σ(2) 1 w σ(3) 1 w σ(4) 1 =0 ( 1) i+j+k+ w w σ() w σ(4) w σ(2) w σ(3) i+w σ() w σ(4) w σ(1) w σ(3) j+w σ() w σ(4) w σ(1) w σ(2) k+w σ() w σ(3) w σ(1) w σ(2) h(w σ()w σ(4) w σ(2) w σ(3) i+w σ() w σ(4) w σ(1) w σ(3) j+w σ() w σ(4) w σ(1) w σ(2) k+w σ() w σ(3) w σ(1) w σ(2) ) E (h) n, w σ(1) w σ(2) w σ(3) w σ(4),w w σ(1) w σ(2) w σ(3) w σ(4) ( w σ() x + w σ() i + w σ() j + w σ() k + w ) σ() w σ(1) w σ(2) w σ(3) w σ(4) hold true for any σ S.

6 14 Ugur Duran et al. It i eaily hown, by uing the definition of x], that y + w x + w k + w ] n n ( ) ( ) n m n w ] = m ] m=0 w 2 w 3 w 4 i + w 1 w 3 w 4 j + w 1 w 2 w 4 k + w 1 w 2 w 3 ] n m w m(w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 ) y + w x] m. (10) Taking w w 1w 2 w 3 w 4 y hw 1w 2 w 3 w 4 y dµ 1 (y) on the both ide of E. (10) yield w w 1w 2 w 3 w 4 y hw 1w 2 w 3 w 4 y y + w x + w k + w ] n n ( ) ( ) n m n w ] = m ] m=0 w 2 w 3 w 4 i + w 1 w 3 w 4 j + w 1 w 2 w 4 k + w 1 w 2 w 3 ] n m w dµ 1 (y) m(w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 ) E (h) m,,w (w x). In view of the E. (11), we acuire (11) ] n ( 1) i+j+k+ (12) =0 w w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 h(w w 4 w 2 w 3 i+w w 4 w 1 w 3 j+w w 4 w 1 w 2 k+w w 3 w 1 w 2 ) w w 1w 2 w 3 w 4 y hw 1w 2 w 3 w 4 y Z p y + w x + w k + w ] n n ( ) n = ] m m w ] n m m=0 dµ 1 (y) E (h) m,,w (w x) U n, w,w w (w 1, w 2, w 3, w 4 m),

7 A Note on Carlitz Twited (h,)-euler Polynomial... 1 where U n,,w (w 1, w 2, w 3, w 4 m) (13) = ( 1) i+j+k+ w w 2w 3 w 4 i+w 1 w 3 w 4 j+w 1 w 2 w 4 k+w 1 w 2 w 3 =0 (m+h)(w 2w 3 w 4 i+w 1 w 3 w 4 j+w 1 w 2 w 4 k+w 1 w 2 w 3 ) w 2 w 3 w 4 i + w 1 w 3 w 4 j + w 1 w 2 w 4 k + w 1 w 2 w 3 ] n m. Hereby, by E. (13), we arrive at the following theorem. Let h Z, w i N be a natural number which atifie the condition w i 1(mod2), in which i Z lie in 1 i. For n 0, the following n m=0 hold true for ome σ S. ( ) n ] m wσ(1)w σ(2)w σ(3)w σ(4) m wσ() ] n m E (h) m, w σ(1) w σ(2) w σ(3) w σ(4),w w σ(1) w σ(2) w σ(3) w σ(4) ( wσ() x ) U n, w σ(),w w σ()(w σ(1), w σ(2), w σ(3), w σ(4) m) Reference 1] E. Ağyüz, M. Acikgoz and S. Araci, A ymmetric identity on the - Genocchi polynomial of higher order under third Dihedral group D 3, Proceeding of the Jangjeon Mathematical Society, 18(2) (201), ] S. Araci, M. Acikgoz and E. Şen, On the extended Kim p-adic - deformed fermionic integral in the p-adic integer ring, Journal of Number Theory, 133(2013), ] S. Araci, U. Duran and M. Acikgoz, Symmetric identitie involving - Frobeniu-Euler polynomial under Sym (), Turkih Journal of Analyi and Number Theory, 3(3) (201), ] U. Duran, M. Acikgoz, A. Ei and S. Araci, Some new ymmetric identitie involving -Genocchi polynomial under S 4, Journal of Mathematical Analyi, 6(4) (201), ] U. Duran, M. Acikgoz and S. Araci, Symmetric identitie involving weighted -Genocchi polynomial under S 4, Proceeding of the Jangjeon Mathematical Society, 18(4) (201), 4-46.

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