Research Article A Note on Symmetric Properties of the Twisted q-bernoulli Polynomials and the Twisted Generalized q-bernoulli Polynomials

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1 Hndaw Publshng Corporaton Advances n Dfference Equatons Volume 00, Artcle ID 80580, 3 pages do:0.55/00/80580 Research Artcle A Note on Symmetrc Propertes of the Twsted q-bernoull Polynomals and the Twsted Generalzed q-bernoull Polynomals L.-C. Jang, H. Y, K. Shvashankara, 3 T. Km, 4 Y. H. Km, 4 and B. Lee 5 Department of Mathematcs and Computer Scence, KonKuk Unversty, Chungu 38-70, Republc of Korea Department of Mathematcs, Kwangwoon Unversty, Seoul 39-70, Republc of Korea 3 Department of Mathematcs, Yuvaraa s College, Unversty of Mysore, Mysore 570# 005, Inda 4 Dvson of General Educaton-Mathematcs, Kwangwoon Unversty, Seoul 39-70, Republc of Korea 5 Department of Wreless Communcatons Engneerng, Kwangwoon Unversty, Seoul 39-70, Republc of Korea Correspondence should be addressed to H. Y, hsy@kw.ac.kr Receved September 009; Revsed 4 Aprl 00; Accepted 3 May 00 Academc Edtor: Abdelkader Boucherf Copyrght q 00 L.-C. Jang et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. We defne the twsted q-bernoull polynomals and the twsted generalzed q-bernoull polynomals attached to χ of hgher order and nvestgate some symmetrc propertes of them. Furthermore, usng these symmetrc propertes of them, we can obtan some relatonshps between twsted q- Bernoull numbers and polynomals and between twsted generalzed q-bernoull numbers and polynomals.. Introducton Let p be a fxed prme number. Throughout ths paper, Q p,andc p wll, respectvely, denote the rng of p-adc ratonal ntegers, the feld of p-adc ratonal numbers, and the completon of algebrac closure of Q p.letv p be the normalzed exponental valuaton of C p wth p p p v pp p. When one talks of q-extenson, q s varously consdered as an ndetermnate, a complex number q C, or a p-adc number q C p.ifq C, one normally assumes q <. If q C p, then we assume q p <p /p, so that q x expx log q for x p cf. 3.

2 Advances n Dfference Equatons For N, d N, weset lm N Z d dp N Z,. see 3. The Bernoull numbers B n and polynomals B n x are defned by the generatng functon as t e t t n B n n!,. t e t ext B n x tn n!.3 cf. 7, 8,, 4, 6. Let UD be the set of unformly dfferentable functons on. For f UD, thep-adc nvarant ntegral on s defned as I f dp N fxdx lm fx. N dp N x0.4 Note that fxdx fxdx see 7. Letf n x be a translaton wth f n x fx n. We note that I n f n I f f 0.5 cf. 3. Km8 studed the symmetrc propertes of the q-bernoull numbers and polynomals as follows: t log q qe t ext Bnx q tn n!..6 In ths paper, we defne the twsted q-bernoull polynomals and the twsted generalzed q-bernoull polynomals attached to χ of hgher order and nvestgate some symmetrc propertes of them. Furthermore, usng these symmetrc propertes of them, we can obtan some relatonshps between the twsted q-bernoull numbers and polynomals and between the twsted generalzed q-bernoull numbers and polynomals attached to χ of hgher order.

3 Advances n Dfference Equatons 3. The Twsted q-bernoull Polynomals Let C p n C p n lm n C p n be the locally constant space, where C p n {ξ ξ pn } s the cyclc group of order p n. For w C p, we denote the locally constant functon by φ w : C p, x w x. cf., 3,, 4. If we take fx φ w xq x e tx, then e xt w x q x dx log q t wqe t.. Now we defne the q-extenson of twsted Bernoull numbers and polynomals as follows: log q t wqe t log q t wqe t etx B q t n n,w Bn,wx q tn n! n!,.3.4 see 3.From.5,.,.3,and.4, we can derve w y q y x y n q dy B n,wx, w y q y y n dy B q n,w..5 By.5, we can see that w nx q nx e nxt dx w x q x e xt dx log q t wn q n e nt w x q x e xt dx t log q wn q n e nt wqe t.6 n w q e t 0 n k0 0 k w q t k k!.

4 4 Advances n Dfference Equatons In.4, tseasytoshowthat n w nx q nx e nxt dx w x q x e xt w x q x e xt dx dx log q t w nx q nx e nxt dx..7 For each nteger k 0, let S q k,w n 0k k wq k w q n k w n q n..8 From.6,.7, and.8, we derve n w nx q nx e nxt dx w x q x e xt w x q x e xt dx dx log q t w nx q nx e nxt dx S q k,w k0 n tk k!..9 From.9, wenotethat B q k,w n Bq k,w ksq k,w n log qsq n, k,w.0 for all k, n N. Letu,u N and w C p ; then we have w u x u x q u x u x e u x u x dx dx w u u x q u u x e u u xt dx t log q wu u q u u e u t w u q u e u t.. By.9,weseethat u w x q x e xt dx w u x q u x e u xt dx l0 u t k l w k q k l l! S q l,w u tl l!.. k0 l0 Let T w u,u ; x, t be as follows: T w u,u ; x, t w u x u x q u x u x e u x u x u u xt dx dx..3 w u u x q u u x e u u xt dx Then we have t log q e u u t w u u q u u e u u t T w u,u ; x, t w u q u e u t w u q u e u t..4

5 Advances n Dfference Equatons 5 From.3, we derve T w u,u ; x, t u w u x q u x e u x u xt dx u w u x q u x e u x t..5 w u u x q u u x e u u xt dx By.4,.,and.5, we can see that T w u,u ; x, t B qu,w u u u x u t S qu! l,w u u ul tl l! 0 l0 n B qu t,w u u xs qu n,w u u u n un n!. 0.6 By the symmetry of p-adc nvarant ntegral on,wealsoseethat T w u,u ; x, t w u x q u x e u x u xt dx u w u x q u x e u x t u w u u x q u u x e u u xt dx n B qu,w u u xs qu t n,w u u u n un n!. 0.7 By comparng the coeffcents of t n /n! on both sdes of.6 and.7,weobtanthe followng theorem. Theorem.. Let u,u,n N. Then for all x, n B qu,w u u xs qu n,w u u u un 0 n 0 B qu,w u u xs qu n,w u u u un,.8 where n s the bnomal coeffcent. From Theorem., f we take u, then we have the followng corollary. Corollary.. For m 0, one we has B q,w u x n B qu,w u xs q n,w u u,.9 0 where n s the bnomal coeffcent.

6 6 Advances n Dfference Equatons By.7,.8, and.9, we can see that e u u xt T w u,u ; x, t w ux q u x e u x t dx u w u x q u x e u x t dx u w u u x q u u x e u u xt dx e u u xt w ux q u x e u x t dx u u w u q u u 0 u 0 B qu n,w u u w u q u e u t 0 w u x q u x e x u xu /u tu dx u x u u n w u q u tn u n!..0 From the symmetry of T w u,u ; x, t, we can also derve T w u,u ; x, t u 0 B qu n,w u u x u u n w u q u tn u n!.. By comparng the coeffcents of t n /n! on both sdes of.0 and., weobtanthe followng theorem. Theorem.3. For m Z, u,u N, we have u 0 B qu n,w u u x u u u n w u q u u 0 B qu n,w u u x u u n w u q u.. u We note that by settng u ntheorem.3, we get the followng multplcaton theorem for the twsted q-bernoull polynomals. Theorem.4. For m Z, u N, one has Bn,wu q u x u n 0 B qu n,w u x w q. u.3 Remark.5. 8, Km suggested open questons related to fndng symmetrc propertes for Carltz q-bernoull numbers. In ths paper, we gve the symmetrc property for q-bernoull numbers n the vewpont to gve the answer of Km s open questons. 3. The Twsted Generalzed Bernoull Polynomals Attached to χ of Hgher Order In ths secton, we consder the generalzed Bernoull numbers and polynomals and then defne the twsted generalzed Bernoull polynomals attached to χ of hgher order by usng

7 Advances n Dfference Equatons 7 multvarate p-adc nvarant ntegrals on.letχ be Drchlet s character wth conductor d N. Then the generalzed Bernoull numbers B n,χ and polynomals B n,χ x attached to χ are defned as t d a0 χaeat e dt t n B n,χ n!, 3. t d a0 χaeat e xt B e dt n,χ x tn n! 3. cf., 8, 3, 7. Let C p n C p n lm n C p n be the locally constant space, where C p n {w w pn } s the cyclc group of order p n. For w C p, we denote the locally constant functon by φ w : C p, x w x 3.3 cf., 3,, 3, 4. If we take fx χxe tx φ w xq x,forq C p wth q p <, then t s obvous from 3. that t log q d χxe tx w x q x a0 dx χawa q a e at. 3.4 w d q d e dt Now we defne the twsted generalzed Bernoull numbers B q n,χ,w and polynomals B q n,χ,wx attached to χ as follows: t log q d a0 χawa q a e at w d q d e dt B q t n n,χ,w n!, 3.5 t log q d a0 χawa q a e at e xt w d q d e dt Bn,χ,wx q tn n! 3.6 for each w C p see 3, 3.By3.5 and 3.6, χxx n w x q x dx Bn,χ,w, q χ y x y n w y q y dy B q n,χ,wx. 3.7

8 8 Advances n Dfference Equatons Thus we have χxe ndxt w nx q nx dx χxe xt w x q x dx log q t nd χxext w x q x dx endxt w ndx q ndx dx wnd q nd e ndt d χe t w q. w d q d e dt Then χxe ndxt w nx q nx dx χxe xt w x q x dx log q t nd nd χle lt w l q l χll k w l q l tk k!. l0 k0 l0 3.9 Let us defne the p-adc twsted q-functon T q χ, n as follows: k,w χ, n χll k w l q l. l0 3.0 By 3.9 and 3.0, weseethat χxe ndxt w ndx q ndx dx χxe xt w x q x dx log q t k0 χ, nd t k k!. 3. Thus, χxnd x k w nx q nx dx χxx k w x q x dx t log q T q k,w χ, nd, 3. for all k, n, d N. Ths means that B q k,χ,w nd Bq n,χ,w t log q χ, nd, 3.3

9 Advances n Dfference Equatons 9 for all k, n, d N. For all u,u,d N, we have d χx χx e w x w x t w u x u x q u x u x dx dx edu u xt w du u x q du u x dx t log q e du u t w du u q du u e du t w du q du e du t w du q du 3.4 d d χae uat w ua q u a χbe ubt w ub q u b. a0 b0 The twsted generalzed Bernoull numbers B k,q n,χ,w and polynomals B k,q n,χ,wx attached to χ of order k are defned as t log q d a0 χawa q a e at k w d q d e dt B k,q t n n,χ,w n!, 3.5 t log q d a0 χawa q a e at k e xt w d q d e dt Bn,χ,wx k,q tn n! 3.6 for each w C p. For u,u N, weset K q w m, χ; u,u d m m χx e m m m x u xu t w m x u xu q m x u xu dx dx m edu u xt w du u x q du u x dx χx e m x u yu t w m x u yu q m x u yu dx dx m, 3.7 where m fx x m dx dx m fx,...,x m dx dx m.in3.7, wenotethat K q wm, χ; u,u s symmetrc n u,u.from3.7, we have K q w m, χ; u,u m m χx e m x u t w m x u q m x u dx dx m e u u xt w u u x q u u x m m χx e m d χx me u x m t w u x m q u x m dx m edu u x q du u x dx x u t w m x u q m x u dx dx m 3.8 e u u yt w u u y q u u y.

10 0 Advances n Dfference Equatons Thus we can obtan u d χxext w x q x dx edu xt w du x q du x dx e u u xt w u u x q u u x m m e u u xt w u u x q u u x k0 u d t χ k w q k k! T q k,w χ, u d tk k!, 0 k0 χx e m x u t w m x u q m x u dx dx m Bn,χ,wu m,q xu n t n n!. d u e du t w du q du χae uat w ua q u a a0 3.9 From 3.9, we derve K q w m, χ; u,u l0 B m,q l,χ,w u xu l l! n 0 t l k0 u un B m,q n,χ,w u x χ, u d tk k! 0 k0 B m,q,χ,w χ, u d k u y u t! B m,q k,χ,w u u y tn n!. 3.0 By the symmetry of K q wm, χ; u,u n u and u, we can see that K q w m, χ; u,u n 0 u un B m,q n,χ,w u x k0 χ, u d k B m,q k,χ,w u y tn 3. n!. By comparng the coeffcents on both sdes of 3.0 and 3., we see the followng theorem. Theorem 3.. For d, u,u,m N, n Z, one has n u un B m,q n,χ,w u x 0 0 k0 n u un B m,q n,χ,w u x χ, u d k k0 B m,q k,χ,w u y χ, u d B m,q k k,χ,w u y. 3.

11 Advances n Dfference Equatons Remark 3.. If we take y 0andm n3., then we have n 0 u un B q n,χ,w u x n 0 u un k0 B q n,χ,w u x χ, u d k k0 χ, u d. k 3.3 K q w Now we can also calculate n m, χ; u,u u k k u n k B m,q n k,χ,w k0 u y du 0 B m,q,χ,w u x u t n u n!. 3.4 From the symmetrc property of K q wm, χ; u,u n u and u, we derve K q w n m, χ; u,u u k k u n k B m,q n k,χ,w k0 u y du 0 B m,q,χ,w u x u t n u n!. 3.5 By comparng the coeffcents on both sdes of 3.4 and 3.6, we obtan the followng theorem. Theorem 3.3. For d, u,u,m N, n Z, we have n u k k u n k B m,q n k,χ,w k0 u y du 0 B m,q k,χ,w n u k k u n k B m,q u y du n k,χ,w k0 0 u x u u B m,q k,χ,w u x u. u 3.6 Remark 3.4. If we take y 0andm n3.6, then one has du u n 0 B q n,χ,w u x u u du u n 0 B q n,χ,w u x u. 3.7 u Remark 3.5. In our results for q, we can also derve smlar results, whch were treated n 7. In ths paper, we used the p-adc ntegrals to derve the symmetrc propertes of the q-bernoull polynomals. By usng the symmetrc propertes of p-adc ntegral on, we can easly derve many nterestng symmetrc propertes related to Bernoull numbers and polynomals.

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