Advanced Studies in Theoretical Physics Vol. 8, 2014, no. 22, HIKARI Ltd,

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1 Advanced Studies in Theoretical Physics Vol. 8, 204, no. 22, HIKARI Ltd, Some Identities of Symmetry for the Higher-order Carlitz Bernoulli Polynomials under Third Dihedral Group D 3 Arising from Volkenborn Integral on Dmitry V. Dolgy Institute of Mathematics and Computer Science Far Eastern Federal University Vladivostok, Russia Seog-Hoon Rim Department of Mathematics Education Kyungpook National University Taegu , Korea Taekyun Kim Department of Mathematics Kwangwoon University, Seoul 39-70, Korea Sang-Hun Lee Division of General Education Kwangwoon University, Seoul 39-70, Korea Copyright c 204 Dmitry V. Dolgy, Seog-Hoon Rim, Taekyun Kim and Sang-Hun 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.

2 978 Dmitry V. Dolgy, Seog-Hoon Rim, Taekyun Kim and Sang-Hun Lee Abstract In this paper, we give some new identities of symmetry for the higher-order Carlitz - Bernoulli polynomials under third Dihedral group D 3 arising from Volkenborn integral on. Mathematics Subject Classification: B68, S40, S80 Keywords: the higher-order Carlitz - Bernoulli polynomial,third Dihedral group D 3, Volkenborn integral. Introduction Let p be a fixed odd prime number. Throughout this paper,, 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 ν p be the normalized exponential valuation of C p with p p p νp(p). Let us assume p that be an indeterminate in C p with p < p p. The analogue number of x is defined as [x] x. Note that lim [x] x. Suppose that f(x) is a uniformly differentiable function on. Then the Volkenborn integral on is defined by Kim to be f(x)dµ (x) lim N lim N p N x0 [p N ] f(x)µ (x + p N ) p N x0 L. Carlitz defined Bernoulli numbers as follows: β 0,, (β + ) n β n, f(x) x, (see [3, 6]). {, if n, 0, if n >, () (2) with the usual convoution about replacing β n by β n, (see [-0]). Recently, Kim gave the Witt s formula for the Carlitz s Bernoulli polynomials, which are given by β n, (x) [x + y] n dµ (y), (n 0), (see [ 0]). (3)

3 Some identities of symmetry 979 When x 0, β n, β n, (0) are Carlitz Bernoulli numbers. By (3), we easily get n ( ) n β n, (x) ( ) l lx l + ( ) n l [l + ] l0 n ( ) (4) n [x] n l lx β l,, (see [3, 6, 9]). l l0 For r N, we consider the higher-order Carlitz Bernoulli polynomials as follows: e t[x+y + +y r] dµ (y ) dµ (y r ) β n,(x) (r) tn n!. (5) Note that where β (r) l, l0 n0 β n,(x) (r) [x + y + + y r ] tdµ (y ) dµ (y r ) n ( ) n ( l + ) r ( ) l lx ( ) n l [l + ] l0 n ( ) n lx β (r) l, l [x]n l, are the higher-order Carlitz s Bernoulli numbers. In this paper, we give some new identities of symmetry for the higher-order Carlitz s Bernoulli polynomials under the third Dihedral group D 3 arising from Volkenborn integral on. Recently, several authors have studied extensions of Bernoulli numbers and polynomials in the several area (see [-6]). (6) 2. Symmetry identities Let w, w 2, w 3 be natural numbers. Then, we have e [w2w3(y+ +yr)+ww2w3x+ww3 i l +w w 2 ( ) r lim N [w p N ] w 2 w 3 w 2w 3 (k l +w y l ). w p N k,,k r0 y,,y r0 [j l ] t dµ w 2 w 3 (y ) dµ w 2 w 3 (y r ) e [w 2w 3 (k l +w y l )+w w 2 w 3 x+w w 3 i l +w w 2 j l ] t (7)

4 980 Dmitry V. Dolgy, Seog-Hoon Rim, Taekyun Kim and Sang-Hun Lee From (7), we can derive the following euation: w 2 w 3 w w 3 i l +w w 2 j l [w 2 w 3 ] r e [w 2w 3 lim N i,,i r0 j,,j r0 y l +w w 2 w 3 x+w w 3 i l +w w 2 [w w 2 w 3 p N ] r w 2 w 3 j l ] t dµ w 2 w 3 (y ) dµ w 2 w 3 (y r ) w p N i,,i r0 j,,j r0 k,,k r0 y,,y r0 y l e [w 2w 3 (k l +w y l )+w w 3 i l +w w 2 j l +w w 2 w 3 x] t. w 2w 3 k l +w w 3 i l +w w 2 j l w w 2 w 3 (8) As this expression is invariant under any permutation σ D 3, we have the following theorem. Theorem. For w, w 2, w 3 N, the following expressions [ w σ(3) ] r e [w σ(3) w σ(3) w σ()w σ(3) i l +w σ() j l i,,i r0 j,,j r0 y l +w σ() w σ(3) x+w σ() w σ(3) i l +w σ() j l ] t dµ w σ(3)(y ) dµ w σ(3)(y r ) are the same for any σ D 3. From (6), we have e [w2w3 [w 2 w 3 ] n n0 y l +w w 2 w 3 x+w w 3 i l +w w 2 [ y l + w x + w w 2 j l ] t dµ w 2 w 3 (y ) dµ w 2 w 3 (y r ) i l + w w 3 j l ] n w 2 w 3 dµ w 2 w 3 (y ) dµ w 2 w 3 (y r ) [w 2 w 3 ] n β (r) n, w 2 w 3 (w x + w (i + + i r ) + w (j + + j r )) tn w n0 2 w 3 n!. (9) Therefore, by Theorem and (9), we obtain the following theorem. Theorem 2. For w, w 2, w 3 N, the following expressions [ w σ(3) ] n [ w σ(3) ] r w σ(3) i,,i r0 j,,j r0 β n, w σ(3) ( w σ() x + w σ() w σ()w σ(3) i l +w σ() j l i l + w σ() w σ(3) ) j l

5 are the same for any σ D 3. Now, we observe that Some identities of symmetry 98 [y + + y r + w x + w (i + + i r ) + w (j + + j r )] w 2 w 3 w 2 w 3 [w ] [w 3 i l + w 2 j l ] w + w w 3 i l +w w 2 j l [ y l + w x] w 2 w 3. [w 2 w 3 ] k0 Thus, by (0), we get [ n ( n k y l + w x + w w 2 ) ( [w ] [w 2 w 3 ] ) n k [w3 i l + w w 3 i l + w 2 (0) j l ] n w 2 w 3 dµ w 2 w 3 (y ) dµ w 2 w 3 (y r ) j l ] n k w k(w w 3 i l +w w 2 From (9), Theorem 2 and (), we can derive the following euation: [w 2 w 3 ] n w 2 w 3 w w 3 i l +w w 2 j l [w 2 w 3 ] r i,,i r0 j,,j Z r0 p [ y l + w x + w i l + w ] n j l dµ w 2 w 3 (y ) dµ w 2 w 3 (y r ) w 2 w 3 w 2 w 3 n ( ) n [w2 w 3 ] k w 2 w 3 [w k [w 2 w 3 ] r ] n k β (r) k, w 2 w 3 (w x) (k+)(w w 3 k0 i,,i r0 j,,j r0 [w 3 (i + + i r ) + w 2 (j + + j r )] n k w n ( ) n [w2 w 3 ] k [w k [w 2 w 3 ] r ] n k β (r) k, w 2 w 3 (w x)r (r) n, w (w 2, w 3 k), k0 where R (r) n,(w, w 2 k) w w 2 i,,i r0 j,,j r0 (k+)(w 2 i l +w j l ) [w2 i l + w j l ) β (r) k, w 2 w 3 (w x). () i l +w w 2 j l ) (2) j l ] n k. (3) As this expression is invariant under third Dihedral group D 3, we have the following theorem. Theorem 3. Let w, w 2, w 3 be natural numbers. Then, for any non-negative integer n, the following expressions n ( ) n [wσ(2) w σ(3) ] k [w k [ w σ(3) ] r σ() ] n k β (r) k, σ(3)(w w σ() x)r (r) n, σ()(w, w σ(3) k), k0

6 982 Dmitry V. Dolgy, Seog-Hoon Rim, Taekyun Kim and Sang-Hun Lee are all the same for σ D 3. ACKNOWLEDGEMENTS. This paper is supported by grant of Russian Scientific Fund. References [] J. Choi, T. Kim, Arithmetic properties for the -Bernoulli numbers and polynorials, Proc. Jangjeon Math. Soc. 5 (202), no.2, [2] J. Choi, T. Kim, Y. H. Kim, A note on the extended -Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math. 2 (20), no. 4, [3] D. V. Dolgy, D.S. Kim, T. Kim, J.-J. Seo, Identities of symmetry for Carlitz -Bernoulli polynomials arising from -Volkenborn integral on g under syymetry group S 3, Advanced Studies in Theoretical physics 8(204), [4] S. Gaboury, R. Tremblay, B.-J. Fugère, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 7 (204), no., [5] D.-J. Kang, S. J. Lee, J.-W. Park, S.-H. Rim, On the twisted weak weight -Bernoulli polynomials and numbers, proc. Jangjeon Math. Soc. 6 (203), no. 2, [6] D. S. Kim, T. K. Kim, -Bernoulri polynomials and -umbral calculus, sci. China Math. 57 (204). no. 9, [7] D. S. Kim, N. Lee, J. Na, K. H. park, Abundant symmetry for higher-order Bernoulli polynomials (II), Proc. Jangjeon Math. Soc. 6 (203), no. 3, [8] D. S. Kim, Symmetry identities for generalized twisted Euler polynomials twisted by unramified roots of unity, Proc. Jangjeon Math. Soc. 5 (202), no. 3, [9] T. Kim, On the weighted -Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math. 2 (20), no. 2, [0] T. Kim, J. Choi, Y.-H. Kim, Some identities on the -Bernstein polynomials, -Stirling numbers and -Bernoulli numbers, Adv. Stud. Contemp. Math. 20 (200), no. 3, [] Q.-M. Luo, F. Qi, Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials, Adv. Stud. Contemp. Math. 7 (2003), no., 8. [2] H. Ozden, I. N. Cangul,Y. Simsek, Remarks on -Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math. 8 (2009), no., 4 48 [3] J.-W. Park, S.-H. Rim, J. Seo, J. Kwon, A note on the modified -Bernoulli polynomials, Proc. Jangjeon Math. Soc. 6 (203), no. 4, [4] J. W. Park, New approach to -Bernoulli polynomials with weight or weak weight, Adv. Stud. Contemp. Math. 24 (204), no.., [5] C. S. Ryoo, T. Kim, A new identities on the.bernoulli numbers and polynomials, Adv. Stud. Contemp. Math. 2l (20), no. 2, 6 69 [6] J.-J. Seo, S.-H. Rim, S.-H. Lee, D. V. Dolgy, T. Kim, -Bernoulli numbers and polynomials related to p-adic invariant integral on, Proc. Jangjeon Math. Soc. 6 (203), no. 3, Received: September, 204