Identities of Symmetry for Generalized Higher-Order q-euler Polynomials under S 3

Transcription

1 Applied Mathematical Sciences, Vol. 8, 204, no. 3, HIKARI Ltd, Identities of Symmetry for Generalized Higher-Order q-euler Polynomials under S 3 Dmitry V. Dolgy Institute of Mathematics and Computer Science Far Eastern Federal University Vladivostok, Russia Yu Seon Jang Department of Applied Mathematics Kangnam University Yongin , Republic of Korea Taekyun Kim Department of Mathematics Kwangwoon University Seoul 39-70, Republic of Korea Jong Jin Seo Department of Applied Mathematics Pukyong National University Pusan , Republic of Korea Copyright c 204 Dmitry V. Dolgy, Yu Seon Jang, Taekyun Kim and Jong Jin Seo. 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 study the identities of symmetry for the generalized higher-order q-euler polynomials in three variable under symmetry group S 3 which are derived from the fermionic p-adic q-integral on Z p.

2 5592 Dmitry V. Dolgy, Yu Seon Jang, Taekyun Kim and Jong Jin Seo. Introduction Let p be a fixed odd prime number. Throughout this paper, Z p, Q p and C p will, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Q p. Let p be the normalized p-adic absolute value with p p /p and let q be an indeterminate in C p such that q < p /p. The q-analogue of x is defined by [x] q q x / q. Note that lim q [x] q x. Let fx be a continuous function on Z p. Then the fermionic p-adic integral on Z p is defined by Kim to be I q f fxdµ q x lim Z N p [p N ] q p N x0 fx q x, where [x] q q x / + q, see [6, 7, 0, ]. For n, by, we get n q n fx + ndµ q x + n fxdµ q x [2] q q l fl n l. Z p Z p l0 2 In particular, for n, we have q fx + dµ q x + fxdµ q x [2] q f0, see [6, 7]. 3 Z p Z p The q-euler polynomials are defined by Kim to be E n,q x [y + x] n q dµ q y, n 0, see [ 8]. 4 Z p When x 0, E n,q E n,q 0 are called the q-euler numbers. For d N with d, p and d mod 2. we set and lim Z/dp N Z, a + dpz p N 0<a<dp a,p a + dp N Z p { x x amod dp N }, where a Z lies 0 a < dp N. Let χ be a primitive Dirichlet character with conductor d N with d mod 2. Then the generalized q-euler polynomials attached to χ are defined by Kim to be χy[x + y] n q dµ q y E n,q,χ [x], n 0. When x 0, E n,q,χ E n,q,χ 0 are called the generalized q-euler numbers attached to χ, see [6, 7].

3 Identities of symmetry 5593 In this paper, we consider the generalized q-euler polynomials attached to χ and study some symmetric identities for the generalized higher-order q-euler polynomials in three variables under symmetry group S 3 which are derived from the fermionic p-adic q-integral on Z p. 2. Symmetry identities for the generalized higher-order q-euler polynomials For r N, let us consider the generalized higher-order q-euler polynomials attached to χ as follows: r χx l e [x + x r+x] qt dµ q x dµ q x r l Thus, by 5, we get E n,q,χx r tn n! n0, see [4, 6, 7]. r χx l [x + x r +x] n q dµ q x dµ q x r E n,q,χx, r n 0. l 6 When x 0, E n,q,χ0 r are called the generalized higher-order q-euler numbers attached to χ. Let w,, w 3 N with w i mod 2, i, 2, 3. Then, by and 5, we get r χx l l e [w 2w 3 l x l+w w 3 x+w w r 3 l i l+w w r 2 l j l] qt dµ q w 3 x dµ q w 3 x l r dp N lim χx N [dp N l ] q w 3 x,,x r0 l e [w 2w 3 l x l+w w 3 x+w w r 3 l i l+w w r 2 l j l] qt q w 3 x + +x r r p N χk [w dp N l ] q w 3 lim N k,,k r0 y,,y r0 l l k l+y l q w 3 l k l+w dy l e [w 3 l k l+ y l +w w 3 x+w w 3 l i l+w l j l] qt, 7 where d N with d mod 2. By 7, we get 5

4 5594 Dmitry V. Dolgy, Yu Seon Jang, Taekyun Kim and Jong Jin Seo [ w 3 ] q r 2 3 i,,i r0 j,,j r0 q w w 3 l i l+w l j l r χi l j l l i l+j l l r χx l e [w 2w 3 l x l+w w 3 x+w w r 3 l i l+w w r 2 l j l] t q dµ q w 3 x dµ q w 3 x r r 2 3 p N lim χi N [ w 3 p N l j l k l ] q l i,,i r0 j,,j r0 k,,k r0 x,,x r0 q w w 3 l i l+w l j l+ w 3 l k l l i l+j l +k l +x l l e [w 3 l k l+ x l +w w 3 x+w w 3 l i l+w l j l] q t q w 3 l x l. 8 As this expression is invariant under any permutation σ S 3, we have the following theorem. Theorem 2.. For w,, w 3, d N with w i mod 2, d mod 2, i, 2, 3, the following expressions r σ2 σ3 r χi l χj l l i l+j l [w σ2 w σ3 ] q i,,i r0 j,,j r0 l l q w σw σ3 l i l+w σ w r σ2 l j l χx l e [w σ2w σ3 l x l+w σ w σ2 w σ3 x+w σ w σ3 l i l+w σ w σ2 l j l] q t dµ q w σ2 w σ3x dµ q w σ2 w σ3x r are the same for any σ S 3. By 6, we get l r χx l e [w 2w 3 l x l+w w 3 x+w w r 3 l i l+w w r 2 l j l] t q l dµ q w 3 x dµ q w 3 x r n0 [ w 3 ] n q E r n,q w 3,χ w x + w l i l + w w 3 j l l t n n!. Therefore, by Theorem and 9, we obtain the following theorem. 9

5 Identities of symmetry 5595 Theorem 2.2. Let w,, w 3, d N with w mod 2, mod 2, w 3 mod 2, d mod 2 and n N {0}. Then the following expressions [ ] n wσ2 w σ2 σ3 σ3 q [ ] r χi l χj l l i l+j l wσ2 w σ3 q i,,i r0 j,,j r l q w σw σ3 l i l+w σ w r σ2 l j l E r n,q w σ2 w w σ3,χ σ x + w σ i l + w σ w σ2 w σ3 are the same for any σ S 3. From 5, we have r [ χx l x l + w x + w l l l i l + w w 3 l ] n j l q w2w3 dµ q w 3 x dµ q w 3 x r [ ] n k n k n [w ] q w 3 i l + j l q kw w 3 l i l+w w r 2 l j l k [ w 3 ] q k0 l l q w r [ ] k χx l x l + w x dµ q w 3 x dµ q w 3 x r l l q w 3 [ ] n k n k n [w ] q w 3 i l + j l q kw w 3 l i l+w w r 2 l j l k [ w 3 ] q k0 E r k,q w 3,χ w x. By 0, we get [ w 3 ] n q [ w 3 ] r q 2 3 i,,i r0 j,,j r0 q w w 3 l i l+w l j l k0 [ l x l + w x + w l l i l+j l l i l + w w 3 l q w l r χi l χj l l r χx l l 0 ] n j l dµ q w 3 x dµ q w 3 x r q w2w3 n [w2 w 3 ] k q [w k [ w 3 ] r ] n k q E r k,q w 3,χ w xt r n,k,q w, w 3 : d χ, q l j l l

6 5596 Dmitry V. Dolgy, Yu Seon Jang, Taekyun Kim and Jong Jin Seo where T r n,k,q w, : d χ r [ χi l χj l l 2 i,,i r0 j,,j r0 i l + w l q w 2 l i l+w l jlk+ l i l+j l ] n k j l. Therefore, by and 2, we obtain the following theorem. l Theorem 2.3. For w,, w 3, d N with w i mod 2, d mod 2, i, 2, 3 and n N {0}, the following expression [ ] k n w σ2 w σ3 q [ ] n k [ ] r wσ E r k q wσ2 w w k,q w σ2 w σ3,χ σxt r n,k,q σw w σ2, w σ3 : d χ σ3 q k0 are the same for any permutation σ S 3. q 2 ACKNOWLEDGEMENTS. This paper is supported by grant of Russian Scientific Fund. References [] S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math , [2] M. Can, M. Cenkci, V. Kurt, and Y. Simsek, Twisted Dedkind type sums associated with Barnes type multiple Frobenius-Euler l-functions, Adv. Stud. Contemp. Math , no. 2, [3] D. Ding and J. Yang, Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials, Adv. Stud. Contemp. Math , 7-2. [4] D. S. Kim and T. Kim, Some identities of symmetry for the generalized q-euler polynomials, Appl. Math. Comput , [5] D. S. Kim, Identities associated with generalized twisted Euler polynomials twisted by ramified roots of unity, Adv. Stud. Contemp. Math , no. 3, [6] T. Kim, q-euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys , no., [7] T. Kim, New approach to q-euler polynomials of higher order, Russ. J. Math. Phys , no. 2, [8] T. Kim, A note on p-adic q-integral on Z p associated with q-euler numbers, Adv. Stud. Contemp. Math , no. 2m [9] T. Kim, Note on the q-euler numbers of higher ordr, Adv. Stud. Contemp. Math , no., [0] T. Kim, Symmetry identities for the twisted generalized Euler polynomials, Adv. Stud. Contemp. Math , no. 2, [] T. Kim, Y. H. Kim, and B. Lee, Note on Carlitz s type q-euler numbers and polynomials, Proc. Jangjeon math. Soc , no. 2,

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