#A51 INTEGERS 14 (2014) MULTI-POLY-BERNOULLI-STAR NUMBERS AND FINITE MULTIPLE ZETA-STAR VALUES

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1 #A5 INTEGERS 4 (24) MULTI-POLY-BERNOULLI-STAR NUMBERS AND FINITE MULTIPLE ZETA-STAR VALUES Kohtaro Imatomi Graduate School of Mathematics, Kyushu Uiversity, Nishi-ku, Fukuoka, Japa k-imatomi@math.kyushu-u.ac.p Received: /27/3, Revised: 5/3/4, Accepted: 7/22/4, Published: /8/4 Abstract We defie the multi-poly-beroulli-star umbers which geeralize classical Beroulli umbers. We study the basic properties for these umbers ad establish sum formulas ad a duality theorem, ad discuss a coectio to the fiite multiple zeta-star values. As a applicatio, we preset alterative proofs of some relatios o the fiite multiple zeta-star values.. Itroductio For ay multi-idex (k,..., k r ) with k i 2 Z, we defie two kids of multi-poly- Beroulli-star umbers B,? (k,...,kr), C,? (k,...,kr) by the followig geeratig series: Li? k,...,k r ( e t ) e t Li? k,...,k r ( e t ) e t B,? (k,...,kr) t!, C,? (k,...,kr) t!, where Li? k,...,k r (z) is the o-strict multiple polylogarithm give by Li? k,...,k r (z) m m r z m m k mkr r Whe r, these umbers are poly-beroulli umbers studied i [], [9]. Further, whe r ad k, both umbers are classical Beroulli umbers sice Li? ( e t ) t. We ote that B,? () C,? () ( 6 ) with B (),? /2 ad C(),? /2. We call k k + + k r the weight of multi-idex (k,..., k r ). We set Q p A : Z/pZ Lp Z/pZ {(a p) p ; a p 2 Z/pZ}/,.

2 INTEGERS: 4 (24) 2 where (a p ) p (b p ) p is equivalet to the equalities a p b p for all but fiitely may primes p. The fiite multiple zeta-star values are defied by? A(k,..., k r ) : H? p (k,..., k r ) mod p p 2 A, where H? (k,..., k r ) is the o-strict multiple harmoic sum defied by H? (k,..., k r ) m m r m k mkr r For more details o the fiite multiple zeta(-star) values, we refer the reader to [7], [] ad [2]. We use star to idicate that the iequalities i the sum are ostrict i cotrast to the strict oes usually adopted i the refereces above. Each of these is expressed as a liear combiatio of the other. This article is orgaized as follows. I 2, we give fudametal properties for the multi-poly-beroulli-star umbers. I 3, we describe the sum formula ad the duality relatio for the multi-poly-beroulli-star umbers. I 4, we study coectios betwee the fiite multiple zeta-star values ad the multi-poly-beroulli-star umbers. As a result, we obtai alterative proofs of some relatios for the fiite multiple zeta-star values.. 2. Basic Properties for the Multi-Poly-Beroulli-Star Numbers I this sectio, we itroduce basic results for the multi-poly-beroulli-star umbers. We first give the recurrece relatios for the multi-poly-beroulli-star umbers B,? (k,...,kr), C,? (k,...,kr). Before statig them, we provide the followig idetity for the o-strict multiple polylogarithm, whose proof is straightforward ad is omitted. Lemma 2.. For ay multi-idex (k,..., k r ) with k i 2 Z, we have d dt Li? k,...,k r (t) 8 >< >: t Li? k,k 2,...,k r (t) (k 6 ), t( t) Li? k 2,...,k r (t) (k, r 6 ), (k r ). t Propositio 2.. For ay multi-idex (k,..., k r ), we have the followig recursios:

3 INTEGERS: 4 (24) 3 (i) Whe k 6, B (k,...,kr),? C (k,...,kr),? + (k,k2,...,kr),k2,...,kr) + B (k,...,kr) C (k,...,kr) A, A. (ii) Whe k, B (,k2,...,kr),? C + + B (k2,...,kr),? where a empty sum is uderstood to be. ( ) B (,k2,...,kr) C (,k2,...,kr) Proof. We prove the relatios for B,? (k,...,kr) : those for C,? (k,...,kr) are similar. Li? k,...,k r ( e t ) ( e t ) B,? (k,...,kr) t!. () We di eretiate both sides of (): Whe k 6, we obtai So we have (LHS) (RHS) e t B,? (k,k2,...,kr) e t e t Li? k,k 2,...,k r ( e t ), B,? (k,...,kr) t! + ( e t ) B,? (k,...,kr) t! (e t ) B (k,...,kr) t +,?!. B (k,...,kr) t +,?! B (k,...,kr) +,? Comparig the coe ciets of t /! o both sides, we obtai (i). Whe k, t A,!. A, (LHS) (RHS) e t Li? k 2,...,k r ( e t ), B (,k2,...,kr) t +,?! + e t B,? (,k2,...,kr) B (,k2,...,kr) t +,?!.

4 INTEGERS: 4 (24) 4 So we have B,? (,k2,...,kr) ad by this we obtai (ii). B (,k2,...,kr) t +,?! e t B (k2,...,kr),? B (k2,...,kr) B (,k2,...,kr) t +,?! B (,k2,...,kr) t +,?! We proceed to describe explicit formulas for the multi-poly-beroulli-star umbers i terms of the Stirlig umbers of the secod kid. We recall that the Stirlig umbers of the secod kid are the itegers for all itegers m, satisfyig m the followig recursios ad the iitial values: + + m (8, m 2 Z), m m m, (, m 6 ). m Propositio 2.2. For ay multi-idex (k,..., k r ), k i 2 Z, we have B,? (k,...,kr) ( ) m+ (m )! ad C (k,...,kr),? + m m r + m m r m k mkr r ( ) m+ (m )! m k mkr r m + m. Proof. Usig the followig idetity (cf. [3, eq. 7.49]) (e t ) m t m! (m ), (2) m! we have B,? (k,...,kr) t! m Li? k,...,k r ( e t ) e t ( e t ) m m m r m k mkr r ( ) m+ (m )! m k mkr r m m m r m t!

5 INTEGERS: 4 (24) 5 + m m r ( ) m+ (m )! m k mkr r m Thus comparig the coe ciets of t /! o both sides, we obtai the explicit formula for B,? (k,...,kr). The explicit formula for C,? (k,...,kr) is obtaied similarly by usig the idetity e t ( e t ) m + ( ) +m+ t (m )! m!, m which follows from (2) by di eretiatio. We fiish this sectio by givig some simple relatios amog the multi-poly- Beroulli-star umbers. Propositio 2.3. For ay multi-idex (k,..., k r ) with k i 2 Z, we have B,? (k,...,kr) C (k,...,kr) ad C (k,...,kr),? ( ) B (k,...,kr). Proof. The geeratig fuctios of B,? (k,...,kr), C,? (k,...,kr) di er by the factor e t, ad the above idetities follow immediately. t!. Propositio 2.4. For ay multi-idex (k,..., k r ), we have C (k,...,kr),? B (k,...,kr) (k,k2,...,kr),? C,?. Proof. By the explicit formula for C,? (k,...,kr), we obtai C (k,...,kr),? + + m m r + m m r + m m r B (k,...,kr) ( ) m+ (m )! + m k mkr r ( ) m+ (m )! m k mkr r (k,k2,...,kr),? C,?. m ( ) m+ (m )! m k m k2 2 mkr r m The secod equality above is by the recursio for the Stirlig umbers of the secod kid. m

6 INTEGERS: 4 (24) 6 3. Mai Results We give the followig sum formulas for multi-poly-beroulli-star umbers. Theorem 3.. We have k + +k rk applerapplek,k i ( ) r B,? (k,...,kr) ( )k k k B () k+,? (3) ad k + +k rk applerapplek,k i ( ) r C,? (k,...,kr) ( )k k k C () k+,?. (4) Proof. We multiply both sides of (3) by t /! ad sums o. Hece we have (LHS) k + +k rk applerapplek,k i k + +k rk applerapplek,k i ( ) r B,? (k,...,kr) t! ( ) r Li? k,...,k r ( e t ) e t, (RHS) ( )k k ( )k k! ( )k k! ( )k k! k B () k+,? B () B,? () t +k! t k e t. ces to prove the followig ide- Sice both sides have the same deomiator, it su tity: k + +k rk applerapplek,k i t k+,? t! ( k + )! ( ) r Li? k,...,k r ( e t ) ( )k t k. (5) k! This equality is proved by iductio o the weight. Whe k, the left-had side is Li? ( e t ) t ad is equal to the right-had side. Next we assume the

7 INTEGERS: 4 (24) 7 idetity holds whe the weight is k. The by di eretiatig the left-had side of the idetity of weight k +, we obtai d dt k + +k rk+ applerapplek+,k i k + +k rk applerapplek,k i k + +k rk applerapplek,k i ( ) r Li? k,...,k r ( e t ) C A e ( ) r t e t e t ( ) r+ Li? k,...,k r ( e t ) Li? k,...,k r ( e t ) ( )k+ t k. k! We used the iductio hypothesis i the last equality. Therefore we have k + +k rk+ applerapplek+,k i ( ) r Li? k,...,k r ( e t ) ( )k+ (k + )! tk+ + C with some costat C, which we fid is by puttig t. The equatio (4) follows from (5) sice the geeratig fuctio of C,? di ers from that of B,? oly by a factor e t. Next we describe the duality relatio for the multi-poly-beroulli-star umbers. We recall the duality operatio of Ho ma [6, p. 65]. We defie a fuctio S from the set of multi-idices (k,..., k r ) with k i ad weight k to the power set of {, 2,..., k } by S((k,..., k r )) {k, k + k 2,..., k + + k r }. Obviously, the map S is a oe-to-oe correspodece. The (k,..., kl ) is said to be the dual idex for (k,..., k r ) i Ho ma s sese whe (k,..., k l) S ({, 2,..., k } S((k,..., k r ))). It is easy to see that Ho ma s duality operatio is a ivolutio. Note that k > if ad oly if k. Theorem 3.2. For ay multi-idex (k,..., k r ) with k i ( apple i apple r), we have C (k,...,kr),? ( ) B (k,...,k l ),?, where (k,..., k l ) is the dual idex of (k,..., k r ) i Ho ma s sese.

8 INTEGERS: 4 (24) 8 Proof. As i the previous proof, cosider the geeratig fuctios of both sides: (LHS) (RHS) C,? (k,...,kr) t! Li? k,...,k r ( e t ) e t, ( ) B (k,...,k l ),? t! Li? ( e t ) k,...,k l e t. Hece we have to show the followig idetity: Li? k,...,k r ( e t ) + Li? k,...,k l( e t ). This idetity also follows from iductio o the weight. First, it is trivial i the case k. Thus, we assume the above idetity holds whe the weight is k. Sice k is equivalet to k 6, we may assume k by the symmetry of the idetity. The whe the weight is k +, the derivative of the left-had side yields Therefore we obtai d Li?,k dt 2,...,k r ( e t ) + Li? ( e t ) k,...,k l e t Li? k 2,...,k r ( e t e t ) + e t Li? ( e t ) k,k 2,...,k l Li? e t k 2,...,k r ( e t ) + Li? ( e t ) k,k 2,...,k l. Li?,k 2,...,k r ( e t ) + Li? k,...,k l ( e t ) C with some costat C, ad by puttig t, we coclude C. 4. Coectio to the Fiite Multiple Zeta-Star Values I this sectio, we give alterative proofs for some relatios of the fiite multiple zeta-star values usig the multi-poly-beroulli-star umbers. The followig cogruece is the star-versio of the cogruece give i [8, Theorem.8], ad is proved i exactly the same maer: Thus we fid H p? (k,k2,...,kr) (k,..., k r ) C p 2,? mod p. A(k?,..., k r ),k2,,kr) p 2,? mod p. (6) p C (k

9 INTEGERS: 4 (24) 9 Corollary 4. (M. Ho ma [7]). For ay multi-idex (k,..., k r ) with k i ( apple i apple r), let (k,..., k l ) be the dual idex for (k,..., k r ) i Ho ma s sese. The we have A(k?,..., k r ) A(k?,..., kl). Proof. It is su ciet to prove the case k. By (6) ad the duality relatio for the multi-poly-beroulli-star umbers, we obtai (,k2,,kr) (LHS) C p 2,? mod p p (RHS) C (k,k 2,,k l ) p 2,? mod p p ( ) p (k2,,kr) B p 2,? mod p (,k2,,kr) Hece we complete the proof if we prove C,? cosider the geeratig fuctios of these umbers: p. (k2,,kr) B,? for all. We C (,k2,,kr) t,? From this we have for ay odd prime p.! Li?,k 2,...,k r ( e t ) e t e t m 2 m r m k2 2 mkr r e t (e t ) Li? k 2,...,k r ( e t ) (k2,,kr) t B,?!. (,k2,,kr) C p 2,? ( ) p B (k2,,kr) p 2,? m m 2 ( e t ) m The followig corollary is a weaker versio of the sum formula for the fiite multiple zeta-star values proved i []. Corollary 4.2. We have k + +k rk r,k 2,k i where B is the classical Beroulli umbers. ( ) r? A(k,..., k r ) (B p k mod p) p,

10 INTEGERS: 4 (24) Proof. Equatios (4) ad (6) yield k + +k rk+ r,k 2,k i ( ) r+? A(k,..., k r ) ( ) k k p 2 C () p k,? k mod p p C () p k,? mod p p. So replacig k by k, we obtai the desired idetity. Ackowledgmets. The author expresses his sicere gratitude to Professor M. Kaeko for useful discussios ad ivaluable advice. Refereces [] T. Arakawa ad M. Kaeko, Multiple zeta values, poly-beroulli umbers, ad related zeta fuctios, Nagoya Math. J. 53 (999), [2] A. Bayad ad Y. Hamahata, Multiple Polylogarithms ad Multi-Poly-Beroulli Polyomials, Fuct. Approx. Commet. Math. 46 (22), [3] R. Graham, D. Kuth, ad O. Patashik, Cocrete Mathematics, Addiso-Wesley, 989. [4] Y. Hamahata ad H. Masubuchi, Special Multi-Poly-Beroulli Numbers, J. Iteger Seq., (27). [5] Y. Hamahata ad H. Masubuchi, Recurrece Formulae for Multi-Poly-Beroulli Numbers, Itegers 7 (27), #A46, 5pp. [6] M. Ho ma, Algebraic aspects of multiple zeta values, Zeta fuctios, Topology ad Quatum Physics, Developmets i Math. 4, Spriger, New York, 25, pp [7] M. Ho ma, Quasi-symmetric fuctios ad mod p multiple harmoic sums, preprit, ariv:math/439v2 [math.nt] 7 Aug. 27. [8] K. Imatomi, M. Kaeko ad E. Takeda, Multi-Poly-Beroulli Numbers ad Fiite Multiple Zeta Values, J. Iteger Seq. 7 (24). [9] M. Kaeko, Poly-Beroulli umbers, J. Théor. Nombres Bordeaux 9 (997), [] M. Kaeko ad D. Zagier, Fiite Multiple Zeta Values, preprit. [] S. Saito ad N. Wakabayashi, Sum formula for fiite multiple zeta values, J. Math. Soc. Japa, to appear. [2] J. Zhao, Wolsteholme type theorem for multiple harmoic sums, It. J. Number Theory 4 (28), 73 6.