Abstract ad Alied Aalysis Volume 2011, Article ID 867217, 9 ages doi:10.1155/2011/867217 Research Article A Note o the Geeralized -Beroulli Measures with Weight T. Kim, 1 S. H. Lee, 1 D. V. Dolgy, 2 ad C. S. Ryoo 3 1 Divisio of Geeral Educatio, Kwagwoo Uiversity, Seoul 139-701, Reublic of Korea 2 Harimwo, Kwagwoo Uiversity, Seoul 139-701, Reublic of Korea 3 Deartmet of Mathematics, Haam Uiversity, Daejeo 306-791, Reublic of Korea Corresodece should be addressed to T. Kim, tkkim@kw.ac.kr Received 21 Aril 2011; Acceted 16 May 2011 Academic Editor: Gabriel Turiici Coyright 2011 T. Kim et al. This is a oe access article distributed uder the Creative Commos Attributio Licese, which ermits urestricted use, distributio, ad reroductio i ay medium, rovided the origial work is roerly cited. We discuss a ew cocet of the -extesio of Beroulli measure. From those measures, we derive some iterestig roerties o the geeralized -Beroulli umbers with weight attached to χ. 1. Itroductio Let be a fixed rime umber. Throughout this aer Z, Q,adC will, resectively, deote the rig of -adic ratioal itegers, the field of -adic ratioal umbers, ad the comletio of algebraic closure of Q.LetN be the set of atural umbers ad Z N {0}.Letν be the ormalized exoetial valuatio of C with ν 1/ see 1 14. Whe we talk of -extesio, is variously cosidered as a idetermiate, a comlex umber C,ora-adic umber C. Throughout this aer we assume that C with 1 < 1, ad we use the otatio of -umber as x 1 x 1, 1.1 see 1 14. Thus, we ote that lim 1 x x. I 2, Carlitz defied a set of umbers ξ k ξ k iductively by ξ 0 1, ( ξ 1 k ξk { 1, if k 1, 0, if k>1, 1.2 with the usual covetio of relacig ξ k by ξ k.
2 Abstract ad Alied Aalysis These umbers are -extesio of ordiary Beroulli umbers B k. But they do ot remai fiite whe 1. So he modified 1.2 as follows: 0, 1, ( 1 1, if k 1, k k, 0, if k>1, 1.3 with the usual covetio of relacig k by k,. The umbers k, are called the k-th Carlitz -Beroulli umbers. I 1, Carlitz also cosidered the exteded Carlitz s -Beroulli umbers as follows: h 0, h, h( k h 1 h k, h 1, if k 1, 0, if k>1, 1.4 with the usual covetio of relacig h k by h k,. Recetly, Kim cosidered -Beroulli umbers, which are differet exteded Carlitz s -Beroulli umbers, as follows: for N ad Z, 0, ( 1, 1,, if 1, 0, if >1, 1.5 with the usual covetio of relacig k by see 3. k, The umbers are called the k-th -Beroulli umbers with weight. k, For fixed d Z with, d 1, we set ( Z d lim, N d N 1 Z, Z ( a dz, 0<a<d a,1 { ( a d N Z x x a mod d N}, 1.6 where a Z satisfies the coditio 0 a<d N. Let UDZ be the sace of uiformly differetiable fuctios o Z. For f UDZ, the -adic -itegral o Z is defied by Kim as follows: I ( f Z fxdμ x lim 1 [ N 1 fx x, N x0 1.7
Abstract ad Alied Aalysis 3 see 3, 4, 15, 16. By1.5 ad 1.7, the Witt s formula for the -Beroulli umbers with weight is give by Z x dμ x,, where Z. 1.8 The -Beroulli olyomials with weight are also defied by,x ( x l l l0 lx l,. 1.9 From 1.7, 1.8,ad1.9, we ca derive the Witt s formula for,x as follows: [ ( x y dμ y,x, where Z. 1.10 Z For Z ad d N, the distributio relatio for the -Beroulli olyomials with weight are kow that,x d d 1 d a a0 ( x a, d d, 1.11 see 3. Recetly, several authors have studied the -adic -Euler ad Beroulli measures o Z see 8, 9, 11, 13, 14. The urose of this aer is to costruct -adic -Beroulli distributio with weight -adic -Beroulli ubouded measure with weight o Z ad to study their itegral reresetatios. Fially, we costruct the geeralized -Beroulli umbers with weight ad ivestigate their roerties related to -adic -L-fuctios. 2. -Adic -Beroulli Distributio with Weight Let be ay comact-oe subset of Q, such as Z or Z.A-adic distributio μ o is defied to be a additive ma from the collectio of comact oe set i to Q : ( μ U k μu k ( additivity, k1 k1 2.1 where {U 1,U 2,...,U } is ay collectio of disjoit comact oesets i. The set Z has a toological basis of comact oe sets of the form a Z. Coseuetly, if U is ay comact oe subset of Z, it ca be writte as a fiite disjoit uio of sets U k ( aj Z, 2.2 j1 where N ad a 1,a 2,...,a k Z with 0 a i < for i 1, 2,...k.
4 Abstract ad Alied Aalysis Ideed, the -adic ball a Z ca be rereseted as the uio of smaller balls 1 a Z (a b 1 Z. 2.3 Lemma 2.1. Every ma μ from the collectio of comact-oe sets i to Q for which b0 μ (a N Z 1 (a b N d N1 Z b0 2.4 holds wheever a N Z, exteds to a -adic distributio ( -adic ubouded measure o. Now we defie a ma μ k, o the balls i Z as follows: μ k, [ ( a k Z [ ( a f {a} k,, 2.5 where {a} is the uiue umber i the set {0, 1, 2,..., 1} such that {a} amod. If a {0, 1, 2,..., 1}, the 1 b0 μ k, [ (a 1 1 k b 1 Z [ b0 1 a [ k [ ( a b ab f k, 1 1 [ k 1 [ b0 b f k, (( a/ b. 2.6 From 2.6, weotethatμ k, is -adic distributio o Z if ad oly if [ k 1 (( a/ [ b f b k, b0 f k, ( a. 2.7 Theorem 2.2. Let N ad k Z. The we see that μ k, a Z is -adic distributio o Z if ad oly if Oe sets [ k 1 (( a/ [ b f b k, b0 f k, ( a. 2.8 f k, k, x x. 2.9
Abstract ad Alied Aalysis 5 From 2.5 ad 2.9, oe gets μ k, [ ( a k Z [ ( a a k,. 2.10 By 1.11, 2.10, adtheorem 2.2, we obtai the followig theorem. Theorem 2.3. Let μ be give by k, [ μ (a d N k d N Z k, [ d N ( a k, dn a d N. 2.11 The μ exteds to a Q-valued distributio o the comact oe sets U. k, From 2.11, oe otes that dμ d N 1 x lim k, lim x0 μ k, [ d N k [ d N (x d N Z d N 1 ( a k, dn a0 a d N. 2.12 By 1.11 ad 2.12, oe gets dμ x k, k,. 2.13 Therefore, we obtai the followig theorem. Theorem 2.4. For N ad k Z, oe has dμ x k, k,. 2.14 Let χ be Dirichlet character with coductor d N. The oe defies the geeralized -Beroulli umbers attached to χ as follows:,χ, χxx dμ x d d 1 a χa d a0, d ( a d. 2.15
6 Abstract ad Alied Aalysis From 2.11 ad 2.15, oe ca derive the followig euatio; χxdμ d N 1 k,x lim x0 [ d N k a0 χxμ k, lim [ d N dk d 1 a χa d lim a0 (x d N Z d N 1 ( χx x k, dn x0 dk d 1 ( a χa a d k, d d [ d N1 k χxdμ x lim [ k, d N1 [ k [ χ ( x d N 1 x0 d k d 1 χ ( a a lim d a0 x d N [ N k [ d N 1 ( a/d x dx k, d N N d x0 k,χ,, ( x x k, dn1 d N1 [ N k [ d N 1 N d x0 dx N ( xd a k, dn d N [ k [ d k d 1 χ ( χa d a0 a k, d ( a d χ ( [ k [ k,χ,. 2.16 For / 1, oe has χxdμ k, 1/ ( x χ ( [ k [ / 1/ ( χxdμ ( 1 k, x χ. 1/ k,χ, 1/ k,χ, /, 2.17 Therefore, we obtai the followig theorem. Theorem 2.5. For / 1, oe has χxdμ k,x k,χ,, χxdμ k, x χ( [ k [ k,χ,,
Abstract ad Alied Aalysis 7 χxdμ k, 1/ ( x χ ( [ k [ / 1/ ( χxdμ ( 1 k, x χ. 1/ k,χ, 1/ k,χ, /, 2.18 Defie μ k,, U μ k,u 1 [ 1 k [ 1 μ k, 1/ ( U. 2.19 By a simle calculatio, oe gets χxdμ k,, x [ 1 k χxdμ k,x 1 [ χxμ x 1 k, 1/ [ 1 k [ χxdμ ( 1 k k, x 1/ k,χ, χ( [ k [, k,χ, [ 1/ k [ 1/ ( χ ( 1 χ [ / k [ / k,χ, 1/ k,χ, /. 2.20 By 2.19 ad 2.20, oe gets χxdμ ( k,, x [ 1 k χxdμ k,x 1 [ χxμ ( 1 k, x 1/ k,χ, χ( [ k [ 1 k,χ, [ 1/ k [ 1/ ( 1 χ k,χ, 1/ 2.21 ( χ [ / k [ / k,χ, /. Now oe defies the oerator χ y χ y,k,: o f by χ y f ( χ y,k,: f ( k χ ( y f ( y. 2.22
8 Abstract ad Alied Aalysis Thus, by 2.22, oe gets χ x,k,: χ y,k,: f ( χ x,k,: k χ ( y f ( y k χ ( y χx k y y χ ( y f ( xy 2.23 [ xy k [ xy χ xy,k,: f ( χ xy f (. χ ( xy f ( xy Let us defie χ x χ y χ x,k,: χ y,k,:. The oe has χ x χ y χ xy. 2.24 From the defiitio of χ x, oe ca easily derive the followig euatio; ( 1 χ ( 1 1 x1/ 1 1 x1/ χ 1 x/. 2.25 Let f. The oe gets k,χ, ( 1 χ ( 1 1 x1/ k,χ, k,χ, 1 1 [ / k [ / [ 1/ k [ 1/ ( χ ( 1 χ [ k k,χ, [ k,χ, /. By 2.21 ad 2.26, oe obtais the followig euatio: χxdμ ( ( k,, x 1 χ ( 1 1 x1/ k,χ,, χ ( k,χ, 2.26 2.27 where / 1. Refereces 1 L. Carlitz, Exasios of -Beroulli umbers, Duke Mathematical Joural, vol. 25,. 355 364, 1958.
Abstract ad Alied Aalysis 9 2 L. Carlitz, -Beroulli umbers ad olyomials, Duke Mathematical Joural, vol. 15,. 987 1000, 1948. 3 T. Kim, O the weighted -Beroulli umbers ad olyomials, Advaced Studies i Cotemorary Mathematics, vol. 21,. 207 215, 2011. 4 T. Kim, O a -aalogue of the -adic log gamma fuctios ad related itegrals, Number Theory, vol. 76, o. 2,. 320 329, 1999. 5 Y.-H.Kim, Othe-adic iterolatio fuctios of the geeralized twisted h, -Euler umbers, Iteratioal Mathematical Aalysis, vol. 3, o. 17 20,. 897 904, 2009. 6 A. Bayad, Modular roerties of ellitic Beroulli ad Euler fuctios, Advaced Studies i Cotemorary Mathematics, vol. 20, o. 3,. 389 401, 2010. 7 A. Bayad, Arithmetical roerties of ellitic Beroulli ad Euler umbers, Iteratioal Algebra, vol. 4, o. 5-8,. 353 372, 2010. 8 H. Ozde, Y. Simsek, S.-H. Rim, ad I. N. Cagul, A ote o -adic -Euler measure, Advaced Studies i Cotemorary Mathematics, vol. 14, o. 2,. 233 239, 2007. 9 H. Ozde, Y. Simsek, ad I. N. Cagul, Euler olyomials associated with -adic -Euler measure, Geeral Mathematics, vol. 15, o. 2,. 24 37, 2007. 10 Y. Simsek, Secial fuctios related to Dedekid-tye DC-sums ad their alicatios, Russia Mathematical Physics, vol. 17, o. 4,. 495 508, 2010. 11 S.-H. Rim ad T. Kim, A ote o -adic Euler measure o Z, Russia Mathematical Physics, vol. 13, o. 3,. 358 361, 2006. 12 S.-H. Rim, J.-H. Ji, E.-J. Moo, ad S.-J. Lee, Some idetities o the -Geocchi olyomials of higher-order ad -Stirlig umbers by the fermioic -adic itegral o Z, Iteratioal Mathematics ad Mathematical Scieces, vol. 2010, Article ID 860280, 14 ages, 2010. 13 N. Koblitz, O Carlitz s -Beroulli umbers, Number Theory, vol. 14, o. 3,. 332 339, 1982. 14 T. Kim, O exlicit formulas of -adic -L-fuctios, Kyushu Mathematics, vol.48,o.1,. 73 86, 1994. 15 T. Kim, -Beroulli umbers ad olyomials associated with Gaussia biomial coefficiets, Russia Mathematical Physics, vol. 15, o. 1,. 51 57, 2008. 16 T. Kim, -Volkebor itegratio, Russia Mathematical Physics, vol. 9, o. 3,. 288 299, 2002.
Advaces i Oeratios Research Advaces i Decisio Scieces Mathematical Problems i Egieerig Algebra Probability ad Statistics The Scietific World Joural Iteratioal Differetial Euatios Submit your mauscrits at Iteratioal Advaces i Combiatorics Mathematical Physics Comlex Aalysis Iteratioal Mathematics ad Mathematical Scieces Stochastic Aalysis Abstract ad Alied Aalysis Iteratioal Mathematics Discrete Dyamics i Nature ad Society Discrete Mathematics Alied Mathematics Fuctio Saces Otimizatio