Symmetric Identities of Generalized (h, q)-euler Polynomials under Third Dihedral Group

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1 Applied Mathematical Sciences, vol. 8, 2014, no. 145, HIKARI Ltd, Symmetric Identities of Generalized (h, )-Euler Polynomials under Third Dihedral Group S. H. Rim Department of Mathematics Education Kyungpook National University Taegu , Republic of Korea T. Kim Department of Mathematics Kwangwoon University, Republic of Korea S. H. Lee Division of General Education, Kwangwoon University Seoul , Republic of Korea Copyright c 2014 S. H. Rim, T. Kim and S. H. Lee. 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 work is properly cited. Abstract In this paper, we give some symmetric identities of generalized (h, )-Euler polynomials under Dihedral group which are derived from p-adic fermionic integral on Z p. 1. Introduction Let p be a fixed odd prime number. Throughout this paper Z p, Q p and C p will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Q p, respectively. The p-adic norm p is normalized as p p 1 p, and the -extension of number x is defined by [x] 1 x 1. Note that lim 1[x] x. Let us assume that is an indeterminate in C p such that 1 p < p 1 p 1. Let f be a continuous function on Zp. Then,

2 7208 S. H. Rim, T. Kim and S. H. Lee the fermionic p-adic integral on Z p is defined by Kim to be I 1 (f) Thus, by (1.1), we get Z p f(x)dµ 1 (x) lim p N 1 x0 f(x)( 1) x. (1.1) n 1 I 1 (f n ) + ( 1) n 1 I 1 (f) 2 f(a)( 1) n 1 a (see [1-16]). (1.2) For d N with (d, p) 1, we get a0 lim N Z dp N Z, where a Z lies 0 a < dp N. Note that f(x)dµ 1 (x) 0<a<dp (a,p)1 a + dpz p, a + dp N Z p { x x a (mod dp N ) }, Z p f(x)dµ 1 (x), (see [9-15]). (1.3) Let χ be a primitive Dirichlet s character with conductor d N with d 1 (mod 2). As is well known, the generalized Euler polynomials attached to χ are defined by χ(y)e (x+y)t dµ 1 (y) 2 d 1 χ(a)e at ( 1) a e xt e dt + 1 a0 E n,χ (x) tn, (see [9]). n! (1.4) Thus, by (1.4), we get χ(y)(x + y) n dµ 1 (y) E n,χ (x), (n 0). (1.5) When x 0, E n,χ E n,χ (0) are called the generalized Euler numbers attached to χ (see [9 10]). In the viewpoint of (1.5), we consider the generalized (h, )-Euler polynomials attached to χ as follows : E n,χ,(x) (h) (h 1)y [x + y] n dµ 1 (y), (see [10]), (1.6) Z p where n N {0} and h Z.

3 Symmetric identities 7209 From (1.6), we can derive the generating function of the generalized (h, )- Euler polynomials attached to χ as follows : χ(y) (h 1)y e [x+y]t dµ 1 (y) E (h) n,χ,(x) tn n!. (1.7) The purpose of this paper is to give the symmetric identities of the generalized (h, )-Euler polynomials under third Dihedral group D 3 which are derived from p-adic fermionic integral on Z p. 2. Some identities of the generalized (h, )-Euler polynomials attached to χ Let h Z and w 1, w 2, w 3 N with w 1 1 (mod 2), w 2 1 (mod 2) and w 3 1 (mod 2). Then we have χ(y)e [w 2w 3 y+w 1 w 2 w 3 x+w 1 w 3 i+w 1 w 2 j] t (h 1)w 2w 3 y dµ 1 (y) dp N 1 lim χ(y)e [w 2w 3 y+w 1 w 2 w 3 x+w 1 w 3 i+w 1 w 2 j] t ( 1) y (h 1)w 2w 3 y lim y0 1 1 P N 1 y0 χ(k) (h 1)w 2w 3 (k+ 1 y) ( 1) k+y e [w 2w 3 (k+ 1 y)+w 1 w 2 w 3 x+w 1 w 3 i+w 1 w 2 j] t, where d N with d 1 (mod2). Thus, by (2.1), we have 2 1 lim 3 1 (h 1)(w 1w 3 i+w 1 w 2 j) χ(i)χ(j)( 1) i+j χ(y)e [w 2w 3 y+w 1 w 2 w 3 x+w 1 w 3 i+w 1 w 2 j] t (h 1)w 2w 3 y dµ 1 (y) p N 1 y0 χ(ijk)( 1) i+j+k+y (2.1) (h 1)(w 1w 3 i+w 1 w 2 j+w 2 w 3 k+ 1 w 2 w 3 y) e [w 2w 3 (k+ 1 y)+w 1 w 2 w 3 x+w 1 w 3 i+w 1 w 2 j] t. (2.2) As this expression is invariant under any permutation σ D 3, we have the following theorem.

4 7210 S. H. Rim, T. Kim and S. H. Lee Theorem expressions 2.1. Let w 1, w 2, w 3, d be odd natural numbers. Then the following σ(2) 1 σ(3) 1 χ(i)χ(j)( 1) i+j (h 1)(w σ(1)w σ(3) i+w σ(1) w σ(2) j) (h 1)w σ(2)w σ(3) y χ(y)e [w σ(2)w σ(3) y+w σ(1) w σ(2) w σ(3) x+w σ(1) w σ(3) i+w σ(1) w σ(2) j] t dµ 1 (y) are the same for any σ D 3. From (1.6), we have (h 1)w 2w 3 y χ(y)e [w 2w 3 y+w 1 w 2 w 3 x+w 1 w 3 i+w 1 w 2 j] t dµ 1 (y) [w 2 w 3 ] n [w 2 w 3 ] n E (h) n,χ, w 2 w 3 [ (h 1)w 2w 3 y χ(y) y + w 1 x + w 1 ( w 1 x + w 1 w 2 i + w 1 w 3 j ) t n w 2 i + w 1 ] n j w 3 w 2 w 3 dµ 1 (y) tn n! n!. (2.3) Therefore, by Theorem 2.1 and (2.3), we obtain the following theorem. Theorem 2.2. For w 1, w 2, w 3, d N with w 1 1 (mod 2), w 2 1 (mod 2) and w 3 1 (mod 2), d 1 (mod 2), the following expressions [w σ(2) w σ(3) ] n E (h) n,χ, w σ(2) w σ(3) σ(2) 1 σ(3) 1 ( 1) i+j χ(i)χ(j) (h 1)(w σ(1)w σ(3) i+w σ(1) w σ(2) j) ( w σ(1) x + w σ(1) w σ(2) i + w σ(1) w σ(3) j are the same for any σ D 3 and n N {0}. By (1.6), we easily see that [ χ(y) (h 1)w 2w 3 y y + w 1 x + w 3 i + w ] n 1 j dµ 1 (y) w 2 w 3 w 2 w 3 n ( ) ( ) n k n [w1 ] [w 3 i + w 2 j] n k k [w 2 w 3 ] w 1 (h 1)w 2w 3 y χ(y)[y + w 1 x] k w 2 w 3 dµ 1 (y) n ( ) ( ) n k n [w1 ] [w 3 i + w 2 j] n k k [w 2 w 3 ] w 1 E (h) k,χ, w 2 w 3 (w 1 x). (2.4) )

5 Thus, by Theorem 2.2 and (2.4), we get, we get where [w 2 w 3 ] n n ( n k Symmetric identities 7211 ( 1) i+j (h 1)(iw 1w 3 +jw 1 w 2 ) χ(i)χ(j) (h 1)w 2w 3 y χ(y) ) T (h) n,k, (w 1, w 2 χ) [ y + w 1 x + w 1 i + w ] n 1 j dµ 1 (y) w 2 w 3 w 2 w 3 [w 2 w 3 ] k [w 1 ] n k E (h) k,χ, w 2 w 3 (w 1 x)t (h) n,k, w 1 (w w, w 3 χ) Therefore, we obtain the following theorem. (2.5) ( 1) i+j (h 1)(w 2i+w 1 j) χ(i)χ(j)[w 3 i + w 1 j] n k. Theorem 2.3. Let w 1, w 2, w 3, d N with w 1 1 (mod 2), w 2 1 (mod 2), w 3 1 (mod 2), d 1 (mod 2) and n N. Then the following expressions n ( ) n [w σ(2) w σ(3) ] k k [w σ(1) ] n k E (h) k, w σ(2) σ(3)(w w σ(1) x)t (h) n,k, σ(1)(w w σ(2), w σ(3) χ) are the same for any σ D 3. References [1] J. Choi, T. Kim, Y. H. Kim and B. Lee,, On the (w, )-Euler numbers and polynomials with weight α, Proc. Jangjeon Math. Soc., 15 (2012), no. 1, [2] J. Choi, T. Kim and Y. H. Kim, A note on the modified -Euler numbers and polynomials with weight, Proc. Jangjeon Math. Soc., 14 (2011), no. 4, [3] S. Gaboury, R. Tremblay and B. J. Fugère, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc., 17 (2014), no. 1, [4] J. H. Jeong, J. H. Jin, J. W. Park and S. H. Rim, On the twisted weak -Euler numbers and polynomials with weight 0, Proc. Jangjeon Math. Soc., 16 (2013), no. 2, [5] D. S. Kim, D. V. Dolgy, T. Kim and S. H. Rim, Identities involving Bernoulli and Euler polynomials arising from Chevyshev polynomials, Proc. Jangjeon Math. Soc., 15 (2012), no. 4, [6] D. S. Kim, Symmetric identities for generalized twisted Euler polynomials twisted by unramified roots of unity, Proc. Jangjeon Math. Soc., 15 (2012), no. 3, [7] T. Kim, On the -extension of higher-order Euler polynomials, Proc. Jangjeon Math. Soc., 15 (2012), no. 3, [8] T. Kim, Y.-H. Kim and B. Lee, Note on Carlitz s type -Euler numbers and polynomials, Proc. Jangjeon Math. Soc. 13 (2010), no. 2, [9] T. Kim, Symmetry properties of the generalized higher-order Euler polynomials, Proc. Jangjeon Math. Soc. 13 (2010), no. 1, [10] T. Kim, A note on the generalized -Euler numbers, Proc. Jangjeon Math. Soc. 12 (2009), no. 1,

6 7212 S. H. Rim, T. Kim and S. H. Lee [11] T. Kim, Symmetry identities for the twisted generalized Euler polynomials, Adv. Stud. Contemp. Math., 18 (2009), no. 2, [12] T. Kim, Y. H. Kim and K. W. Hwang, On the -extension of the Bernoulli and Euler numbers related identities and Lerch zeta function, Proc. Jangjeon Math. Soc., 12 (2009), no. 1, [13] Y.-H. Kim, W. Kim and C.S. Ryoo, On the twisted -Euler zeta function associated with twisted -Euler numbers, Proc. Jangjeon Math. Soc., 12 (2009), no. 1, [14] Q. M. Luo and F. Qi,, Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials, Adv. Stud. Contemp. Math., 7 (2003), no. 1, [15] C. S. Ryoo, On the generalized Barnes type multiple -Euler polynomials twisted by ramified roots of unity, Proc. Jangjeon Math. Soc., 13 (2010), no. 2, [16] Z. Zhang and H. Yang, Some closed formulas for generalized Bernoulli-Euler numbers and polynomials, Proc. Jangjeon Math. Soc. 11 (2008), no. 2, Received: September 3, 2014