Poly-Bernoulli numbers and related zeta functions

Transcription

1 Poly-Beroulli umbers ad related zeta fuctios Masaobu Kaeko Itroductio I this expository article, we review some aspects of poly-beroulli umbers ad related zeta fuctios. The poly-beroulli umber is a geeralizatio of the classical Beroulli umber usig the polylogarithm series. Although its defiitio looks rather artificial at first glace, it has tured out recetly that the poly-beroulli umbers of egative idex have very ice combiatorial iterpretatios, ad also they appear i special values of certai zeta fuctios. It may therefore be reasoable to seek arithmetic properties that may be ivolved with poly-beroulli umbers. The author made oe such attempt with late Arakawa i the hope of fidig a ice zeta fuctio which coects poly-beroulli umbers with the so-called multiple zeta values, the subject of wide iterest ot oly i umber theory but also i umerous other braches such as topology, quatum groups, arithmetic geometry, mathematical physics etc. This work with Arakawa will be reviewed i 3, after recallig defiitios ad properties of poly-beroulli umbers i 2. I 4 we give some results ad speculatios cocerig the multiple harmoic sums mod p ad multiple zeta-star values. I the fial sectio, 5, we discuss a differet type of zeta fuctio which also has some relatio to poly-beroulli umbers as well as to certai geeralized multiple zeta values. The author would like to take this opportuity to express his deep gratitude to late Professor Tsueo Arakawa o the occasio of his sixtieth birthday, whose ecouragemet ad iterest at the early stage of the research o this topic greatly helped i developig the work further. 2 Poly-Beroulli umbers I relatio to the well-kow formula for the sum of cosecutive powers of itegers, Takakazu Seki 2 ad Jacob Beroulli 3 idepedetly itroduced a sequece of ratioal umbers, owadays kow as the Beroulli umbers, B ( = This is a exteded versio of [20]. 2 Katsu-you-sa-pou, published posthumously i Ars cojectadi, published also posthumously i 73.

2 0,, 2,...). Their defiitio is by the recursio ( ) + B i = + ( = 0,, 2,...), i i=0 ad this ca be expressed by meas of a geeratig series as xe x e x = x B!. =0 We ote that the left-had side of this differs by x from the more commoly used defiig series; xe x e x = x e x + x, ad as a result, with our defiitio we have B = /2 (istead of /2). The other values of B are the same i both defiitios ad, because B = 0 for odd 3, to covert ay formula with oe defiitio ito the other we oly eed to chage B ito ( ) B. Accordig to [8] ad [4], we defie the poly-beroulli umber B (k) ad its relative C (k), for ay itegers k Z ad 0, by the geeratig series Li k ( e x ) e x = =0 B (k) x! ad Li k ( e x ) e x = =0 C (k) x! respectively 4. Here, Li k (z) deotes the formal power series m= zm /m k (the kth polylogarithm whe k > 0, ad the ratioal fuctio (z d/dz) k (z/( z)) whe k 0). Whe k =, we have Li (z) = log( z) ad these geeratig series become xe x e x ad x e x respectively, ad hece each of B (k) geeralizes the classical Beroulli umbers B, by choosig oe of the above geeratig series for B. Sice the two geeratig series for B (k) Also, usig B (k) = m=0 ( ) C m (k), m ad C (k) ad C (k) C (k) Li k ( e x ) e x = e x Li k( e x ) e x = Li k( e x ) e x differ by a factor e x, the two umbers = ( ) x + 0 ( ) ( ) m B (k) m. m m=0 = Li k( e x ) e x + Li k ( e x ) Li ( e x ) e x dx, 4 We use the otatio B (k) istead of B (k), to avoid possible cofusio with Carlitz s Beroulli umber of higher order. 2

3 we have the relatio B (k) = C (k) + C (). I particular, specializig k = 2 ad usig the fact that C () = 0 for odd 3, we have B (2) = C (2) for eve 4. () We review here some of the kow properties of poly-beroulli umbers. The first is the closed formulas i terms of the Stirlig umber of the secod kid. The Stirlig umber of the secod kid, deoted by { i}, is the umber of ways to partitio a set of elemets ito i oempty subsets. Theorem We have the followig formulas: ) For ay k Z ad 0, B (k) = ( ) 2) For k, 0, i=0 ( ) i i! { } i (i + ) k, C (k) = ( ) i=0 ( ) i i! { } + i+ (i + ) k. B ( k) = C ( ) = mi(,k) mi(,k) { }{ } + k + (j!) 2, j + j + { + j!(j + )! j + }{ k + j + }. Proof. To prove ), we expad the defiig geeratig series by usig the formula (see e.g. [7]) { } x (e x ) i = i! (2) i! =i for the Stirlig umbers ad compare the coefficiets. For 2), we calculate the two variable geeratig series =0 k=0 B ( k) x y k! k! ad =0 k=0 C ( ) x y k! k! by usig formulas i ), ad as a result we have e x+y e x + e y e x+y ad e x+y (e x + e y e x+y ) 2 3

4 respectively. Writig the first expressio as e x+y e x + e y e x+y = e x+y (e x )(e y ) = e x+y (e x ) j (e y ) j = ad usig (2), we obtai the first formula of 2). similarly by usig e x+y (e x + e y e x+y ) 2 = (j + ) 2 d dx (ex ) j+ d dy (ey ) j+ e x+y ( (ex )(e y ) ) 2 = e x+y (j + )(e x ) j (e y ) j = The secod ca be proved d (j + ) dx (ex ) j+ d dy (ey ) j+. The { mi(, k) i the upper limits i the formulas is because the Stirlig umber } k is 0 whe < k. Corollary For k, 0, we have the symmetries B ( k) = B ( ) k ad C ( ) = C ( ) k. With the aid of the above explicit formulas, C. Brewbaker [8, 9] ad S. Lauois [22] foud beautiful combiatorial iterpretatios of B ( k), which we ow describe briefly. A loesum matrix is a matrix with etries 0 ad whose row-sums ad ) colum-sums determie the matrix uiquely. For istace, the matrix ( 0 0 gives t (, 2, ) ad (3, ) as row- ad colum-sums respectively, ad from these two vectors, the origial matrix is recovered uiquely. The theorem of Brewbaker states that the umber of loesum matrices of a give size is equal to the poly-beroulli umber. Theorem (Brewbaker [8, 9]) matrices is equal to B ( k). For k,, the umber of k loesum The key fact i order to cout the total umber of loesum matrices is the characterizatio (usig a old result of Ryser [27]) to the effect that a (0, ) )- matrix is loesum if ad oly if it has o 2 2 mior of the form or ( 0 0 4

5 ( ) 0 0. For a complete proof, we refer the reader to the origial papers [8, 9]. The secod combiatorial iterpretatio of B ( k) is related to the umber of special type of permutatios. Let S deote the symmetric group of order, idetified with the set of all permutatios o the set {, 2,..., }. Lauois proved the followig. Theorem (Lauois [22]) Let k ad be positive itegers. The cardiality of the set {σ S k+ k σ(i) i, i k + } is equal to B ( k). We omit the proof ad oly refer to [22]. It may be a iterestig problem to establish a atural bijectio betwee the sets of k loesum matrices ad the above permutatios. We ote that either of these iterpretatios of B ( k) makes the above duality formula B ( k) = B ( ) k apparet. Further results obtaied i [8, 5, 9] iclude Clause-vo Staudt type theorems for B (k), ad a aalogue for C (k) of the Akiyama-Taigawa algorithm for computig Beroulli umbers (similar to Pascal s triagle for biomial coefficiets). As for the Clause-vo Staudt type result, a complete descriptio of deomiators of B (2) (di-beroulli umbers) is give i [8] ad partial results are obtaied i [5] for geeral k. A importat ope problem is to fid a Kummer type cogruece for poly- Beroulli umbers. This ad its geeralizatio may be of importace also i the theory of p-adic multiple zeta values, as surmised by Furusho [3]. As aother topic of further ivestigatio, we poit out that the extra symmetries or other ice properties of dilogarithm fuctio (see [32]) may force di-beroulli umbers (the case of the upper idex k = 2) to have the more rich properties tha the other oes (k 2). We also poit out that i the di-beroulli case, both umbers B (2) ad C (2) coicide whe is eve, as oted before (). 3 Multiple zeta values ad a zeta fuctio The multiple zeta value (MZV) is a real umber associated to each idex set (k, k 2,..., k ) of positive itegers with k 2, defied by the coverget series ζ(k, k 2,..., k ) := m >m 2 > >m >0 m k mk 2 2. mk This is a rather aive geeralizatio of values of the Riema zeta fuctio ζ(s) at positive iteger argumets, whose study was iitiated by Euler [2] i the case of depth = 2. Sice 990 s whe coectios to quatum field theory, 5

6 kot theory, mixed Tate motive, or quatum groups were foud ([0], [23], [4], []), the MZV has become a topic of itesive study. We refer the iterested reader to Mike Hoffma s web page [6] for extesive refereces o MZV s. I [4], we studied the fuctio ξ k (s) (k ) defied by ξ k (s) := t s Γ(s) 0 e t Li k( e t )dt, (3) with the itetio that we might be able to fid a geeralizatio of Euler s celebrated formulas B 2k ζ(2k) = ( ) 2(2k)! (2π)2k, ζ( k) = B k k (k ), with multiple zeta values o oe had, ad poly-beroulli umbers o the other. What we obtaied i [4] is the followig theorem. Theorem 2 ([4]) ) The itegral (3) coverges for Re(s) > 0 ad the fuctio ξ k (s) aalytically cotiues to a etire fuctio of s. 2) We have the relatio ξ k (s) = ( ) [ ζ(s, 2,,..., ) + ζ(s,, 2,,..., ) + + ζ(s,,...,, 2) + s ζ(s +,,..., ) ] k 2 + ( ) j ζ(k j) ζ(s,,..., ). (4) j Here, ζ(s, k 2,..., k ) is a oe variable fuctio i s with fixed k 2,..., k, cotiued meromorphically to the whole s-plae. I particular, the values of ξ k (s) at positive itegers are writte i terms of MZV s. 3) At o-positive iteger argumets, we have ξ k ( ) = ( ) C (k) ( = 0,, 2,...). (5) We oly briefly metio to the proof. Oce the holomorphic cotiuatio is established i a stadard maer such as used i [33, 4 of Part ], the relatio (4) is obtaied by computig the itegral 0 0 x s k x + + x k e x + +x k e x 2+ +x k e x k dx dx k i two ways, oe is by the repeated use of the Melli trasform ad the other is the itegratio by parts usig x Li 2 ( e x x k ) = x + + x k e x + +x k 6

7 ad similar formulas for higher Li k. Formula (5) i 3) is deduced also i a stadard way. Or rather, we have so defied the fuctio ξ k (s) that we have (5). (Note, however, we face with a covergece problem if we wat B (k) istead of C (k).) Iterestig poit is that the fuctio ξ k (s) has the expressio (4) i terms of the multiple zeta fuctio. We remark that the multi-variable fuctio ζ(s, s 2,..., s ) := m >m 2> >m >0 m s ms 2 2 ms is also meromorphically cotiued to C, thaks to the works of Akiyama- Egami-Taigawa [2] ad Zhao [34]. To seek for a coectio betwee poly- Beroulli umbers (or its geeralizatio) ad the values of ζ(s, s 2,..., s ) at o-positive itegers may be a iterestig problem, but, as described i [2], those poits are poits of idetermiacy ad we have o caoical values there. Still, it is possible to fid a coectio with ay fixed way of limitig process. As for values at positive itegers of ξ k (s), formulas ad ξ k () = a + +a k = a j 0 (a + )ζ(a + 2, a 2 +,..., a k + ) (k, ), ξ k (2) = k 2 ( ) i ζ(i + 2)ζ(k i) (k: eve 2) 2 i=0 are obtaied i [4]. Y. Oho discovered, as a applicatio of his reowed formula [25], that the first expressio ca be trasformed ito the followig simple formula: Theorem (Oho [25]) For k,, we have ξ k () = ζ (k +,,..., ), (6) where ζ (k, k 2,..., k ) := is the multiple zeta-star value. m m 2 m m k mk 2 2 mk This formula plays a iterestig role i fidig a duality pheomeo of multiple zeta-star values, which we discuss i the ext sectio. 7

8 4 Fiite multiple zeta sums mod p ad multiple zeta-star values Let p be a odd prime umber. Cosider the fiite sums obtaied by trucatig the series for ζ(k, k 2,..., k ) ad ζ (k, k 2,..., k ) right before the prime p appears i deomiators; H p (k, k 2,..., k ) := H p (k, k 2,..., k ) := p m >m 2> >m p m m 2 m m k mk 2 2, mk m k mk 2 2. mk Hoffma [5] ad Zhao [35] studied these sums ( multiple harmoic sums i Hoffma s termiology) mod p. For a particular type of idex ( height case), Hoffma showed that there is essetially o differece i modulo p betwee zetas ad zeta-stars : Theorem (Hoffma [5, Th. 5.]) For prime p >, it holds the cogruece Hp (k,,..., ) ( ) k H p (k,,..., ) mod p. Ad he cojectures that all sums H p (k, k 2,..., k ) ad Hp (k, k 2,..., k ) ca be writte mod p as sums of products of the height oe sums H p (k,,..., ). For these cojectural buildig blocks H p (k,,..., ) mod p, he gave a closed formula as a sum ivolvig the Stirlig umbers of the secod kid. A simple maipulatio usig Fermat s little theorem ad the closed formula of i Theorem -2) shows that his formula ca be stated simply i terms of poly-beroulli umbers as follows. C (k) Theorem (Hoffma [5, Th. 5.4]) For k, ad ay prime p >, we have Hp (k,,..., ) ( ) C () p mod p. (7) Proof. The formula of Hoffma [5, Th. 5.4] reads p { } p H p (k,,..., ) ( ) j ( j) p k (j )! mod p. j j= By Fermat s little theorem, j p k j k mod p, the right-had side is cogruet 8

9 to p ( ) p k j= p ( ) k+ ( ) p ( ) j (j )! { } p j j mod p ( ) k+ C () p mod p (by Theorem ). ( ) j j! { } p j+ (j + ) mod p (j j + ) This cogruece, combied with the previous theorem of Hoffma, establishes the theorem. With (7) ad (5) together, we obtai Hp (k,,..., ) ξ ( p + + ) mod p. (8) I view of the formula (6) of Oho, this cogruece looks very suggestive, although for the momet it is oly a superficial curiosity. The curious poit is this: Start with the value of the fuctio ξ (s) at a positive iteger. This is, by Oho s Theorem (6), the multiple zeta-star value ζ (k,,..., ). Take a odd prime p ad trucate this series to get Hp (k,,..., ), ad reduce it modulo p. The the resultig value is cogruet mod p to the value of ξ (s) at (p ), the shift of the iitial by p! ξ () = ζ (k,,..., ) trucate = Hp (k,,..., ) mod p = ξ ( (p )). Now, we trace our origial thikig to get the idea of a kid of duality for multiple zeta-stars. I the same paper, Hoffma also proved the duality cogruece ([5, Th.5.2]) ( ) Hp (,,..., ) ( ) k Hp (k,,..., ) mod p. Give the relatio to poly-beroulli umbers (7), this is just a cosequece of the duality of poly-beroulli umbers i Corollary to Theorem. As a possible differet approach however, first ote the cogruece of trucated Riema zeta values (p ) 0 mod p which is valid for all p > +. This is because the left-had side is equal to (p ) = p(p )/2 0 mod p. The above metioed duality of Hoffma would follow from this if the differece ( ) Hp (,,..., ) ( ) k Hp (k,,..., ) 9

10 could be expressed as a polyomial i the trucated Riema zeta values. Actually, Hoffma proved i [5] may of the cogrueces i this way. However, the duality i questio is proved i aother way ad we do ot kow if there is such a expressio. Ayway, ispired by this ad the above metioed curious aalogy, we surmised that the differece of the two multiple zeta-star values ( ) ζ (,,..., ) ( ) k ζ (k,,..., ) (9) may be writte as a polyomial over Q i the Riema zeta values, ad did umerical experimets. The result was i favor of the speculatio, ad soo after the author had iformed him of this speculatio, Yasuo Oho proved that this was ideed true. He obtaied, usig (6) ad (4) together with his mai result i [25], the formula ( ) ζ (,,..., ) ( ) k ζ (k,,..., ) = (k )ζ(k +,,..., ) ( )ζ( +,,..., ) 2 k 2 k 2 +( ) k ( ) j ζ(k j)ζ(,,..., ) j= j 2 ( ) ( ) j ζ( j)ζ(k,,..., ). j= j Sice we kow that the multiple zeta values of height (= of type ζ(m,,..., )) ca be expressed as polyomials over Q i the Riema zeta values ([3], [], see also [26]), we coclude that the quatity (9) is a polyomial i the Riema zeta values. Usig this formula, we ca compute the two variable geeratig series of (9): ( ( ) ζ (,,..., ) ( ) k ζ (k,,..., ) ) x y k, 2 Γ( x)γ( y) = ψ(x) ψ(y) + π (cot(πx) cot(πy)). Γ( x y) Here, ψ(x) = Γ (x)/γ(x) is the digamma fuctio. To compute this, we use the formula of Aomoto [] ad Drifeld [] k, ζ(k +,,..., )x k y = Γ( x)γ( y) Γ( x y) ad the well-kow Taylor expasio of the (logarithm of) gamma fuctio ( Γ( + x) = exp γx + ( ) ζ() ) x ( x <, γ : Euler s costat). =2 0

11 From this we have Γ( x)γ( y) Γ( x y) = exp ( =2 ) ζ() (x + y (x + y) ), ψ(x) = x γ + ( ) ζ()x, =2 π cot(πx) = x + ψ( x) ψ( + x) = x 2 ζ(2)x 2, ad expadig these out we obtai a rather complicated (i fact too complicated to eatly describe, because we have to expad the expoetial) expressio of ( ) ζ (,,..., ) ( ) k ζ (k,,..., ) as a polyomial i Riema zeta values. All the details ad possible geeralizatios will be discussed i a joit paper [2]. Recall the duality (i height case) for the usual multiple zeta values; ζ( +,,..., ) = ζ(k +,,..., ). This does ot hold for ζ -values whe we just replace ζ by ζ, ad, to the best of our kowledge, o duality-like formula for ζ is kow so far. The established assertio = ( ) ζ (,,..., ) ( ) k ζ (k,,..., ) Q[ζ(2), ζ(3), ζ(5),...] may be regarded as a kid of duality (modulo the rig of Riema zeta values Q[ζ(2), ζ(3), ζ(5),...]). We do ot kow the reaso why the correspodece of idices (,,..., ) (k,,..., ) for this ζ case is differet from that of the duality of usual multiple zeta values, ( +,,..., ) (k +,,..., ) Fially, we poit out the potetial importace of studyig further the fuctio ξ(k,..., k r ; s), a multiple geeralizatio of ξ k (s) itroduced i [4], i order to uderstad ad geeralize properties ad pheomea of the multiple zeta-star values discussed i this sectio.

12 5 Values of the cetral biomial series I this fial sectio, we review some facts o the values of the cetral biomial series ζ CB (s), defied by the followig absolutely coverget Dirichlet series; ζ CB (s) := m s( ) 2m ( s C). m m= I [6], Borwei, Broadhurst ad Kamitzer show that the value ζ CB (k) for each positive iteger k 2 is writte as a Q-liear combiatio of multiple zeta values (of height ) ad multiple Clause ad Glaisher values. The latter two are real or imagiary parts (accordig to the parity of weights) of values at a 6th root of uity of the multiple polylogarithm fuctio Li k,...,k (z) := m > >m >0 z m m k. mk I aalogy with Zagier s cojecture for multiple zeta values 5, they cojecture the followig Cojecture ([6]) Cosider the followig dimesios of the Q-vector spaces ( ) a k := dim Q Q Re i k+ +k Li k,...,k (e πi/3 ), b k := dim Q k + +k=k k i, k + +k=k k i, ( ) Q Im i k + +k Li k,...,k (e πi/3 ). The, these umbers a k ad b k are determied recursively by a 0 = a =, b 0 = b = 0, a = a + b 2, b = b + a 2. I particular, the umber a + b is the Fiboacci umber. It would be a very iterestig problem to fid a arithmetic/geometric iterpretatio of the cojecture ad to prove, as i Gocharov [4] ad Terasoma [30], that these umbers actually give upper bouds of the spaces. O the other had, all the values ζ CB (k) for k are Q-liear combiatios of ad π/ 3. This fact follows from a result due to D. H. Lehmer [24], who adopted the formula 2x arcsi(x) x 2 = (2x) 2m m ( ) 2m x < m= m 5 This is a cojecture posed i [3] cocerig the dimesio of the Q-vector space spaed by MZV s of fixed weight. The cojecture predicts that the dimesios i questio satisfy a simple Fiboacci-like recursio. Decisive result to the effect that the cojectural dimesio does give a upper boud was give by Gocharov [4] ad Terasoma [30]. 2

13 ad its successive differetiatios to derive the followig explicit formula. Defie two sequeces of polyomials {p k (t)} ad {q k (t)} (k =, 0,, 2,...) over Z by p (t) = 0, q (t) = ad the recursio p k+ (t) = 2(kt + )p k (t) + 2t( t)p k(t) + q k (t) (k ), q k+ (t) = (2(k + )t + ) q k (t) + 2t( t)q k(t) (k ). The first few examples are m= p 0 (t) =, p (t) = 3, p 2 (t) = 8t + 7, p 3 (t) = 20t t + 5,..., q 0 (t) =, q (t) = 2t +, q 2 (t) = 4t 2 + 0t +,.... The we have for k (2m) k (2x) 2m x ( ( 2m ) = x ) x ( x 2 ) 2 p k+3/2 k (x 2 ) + arcsi(x)q k (x 2 ) m ad cosequetly ζ CB ( k) = 3 ( 2 3 ) k ( ) p k ( 2 3 ) k+ ( ) π q k 4 3 (k ). (0) This shows that the values ζ CB (k) (k ) all lie i the two dimesioal Q- vector space spaed by ad π/ 3, the fact which is parallel to the result of Euler for ζ(s): Namely, the values of ζ(s) at positive itegers give variety of (cojecturally idepedet) trascedetal umbers icludig powers of π (at eve argumets) ad almost ukow ζ(odd), whereas the values at egative itegers all lie i the oe dimesioal Q-vector space, Q itself, ad these values are explicitly described by the Beroulli umbers. It is therefore iterestig to ote that, for the value (0), R. Stepha [29] observed the (still cojectural) formula ( ) k ( ) 2 p k = 3 4 k B ( j) k j. A explicit formula give i [7] may be of help to establish this idetity. It would also be iterestig if we could fid ay coectio of the coefficiet of π/ 3 i (0) to poly-beroulli or allied umbers, but so far o such coectio seems to have bee foud. We may cosider various aalogues of the fuctio ζ CB (s) ad its values at iteger argumets. It is possible that amog them there are similar descriptios as i the case of ζ CB (s) described above. Refereces [] A. Adelberg, Kummer cogrueces for uiversal Beroulli umbers ad related cogrueces for poly-beroulli umbers, It. Math. J,, (2002). 3

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16 [34] J. Zhao, Aalytic cotiuatio of multiple zeta fuctios, Proc. Amer. Math. Soc., , (2000). [35] J. Zhao, Wolsteholme type theorem for multiple harmoic sums, It. J. Number Theory, 4 o., 73 06, (2008). Faculty of Mathematics, Kyushu Uiversity 33, Fukuoka , Japa. mkaeko@math.kyushu-u.ac.jp 6