Zeta stars. Yasuo Ohno and Wadim Zudilin

Transcription

1 counications in nuber theory and physics Volue 2, Nuber 2, , 2008 Zeta stars Yasuo Ohno and Wadi Zudilin We present two new failies of identities for the ultiple zeta star values: The first one generalizes the forula ζ {2} n, 2ζ2n +, where {2} n denotes the n-tuple 2, 2,...,2, while the second faily is a weighted analogue of Euler s forula n l2 ζl, n l ζn n 3. Introduction Although the structure of algebraic relations of the ultiple zeta values MZVs ζs ζs,...,s n, s,...,s n {, 2,...}, s 2, a s asn n a > >a n got a conjectural description by eans of shuffle and stuffle, or haronic, relations and there was a big progress on obtaining the expected upper bounds for diensions of the Q-spaces spanned by MZVs of fixed weight s wts s + + s n recent works of Deligne and Goncharov [3] and Terasoa [6], new paraetric failies of elegant identities for MZVs continue to coe. It is worth entioning the deep relations between MZVs and quantu physics, knot theory, Galois representations of the fundaental group of P \{0,, }, connection forulae of differential equations and so on. For exaple, calculating counter-ters of Feynan diagras usually gives us the MZVs, and this is a natural link between quantu field theory, nuber theory and knot theory see [2, 2]. The iterated integral representation of MZVs originates fro the Knizhnik Zaolodchikov KZ equation which appears in conforal two-diensional quantu field theory. The foralized KZ equation is solved in ters of the iterated integral, and the ratio of two of its solutions turns out to be a generating series of MZVs which is called the Drinfeld associator [4, ]. 325

2 326 Yasuo Ohno and Wadi Zudilin Many of MZV identities have a siple for for an alternative odel of zeta values, so-called ultiple zeta star values MZSVs or zeta stars, 2 ζ s ζ s,...,s n, s,...,s n {, 2,...}, s 2, a s...asn n a a n see the discussion in K. Ihara, J. Kajikawa, Y. Ohno and J. Okuda, MZSVs vs. MZVs, in preparation. The equivalence of the odels of MZVs and MZSVs 2, in the sense that the Q-spaces in R spanned by MZVs and by MZSVs of fixed weight coincide, is a well-known fact it is an iediate consequence of Proposition below. In our further discussion, we require soe standard characteristics of the index s in and 2: the already defined weight s and the depth or the length ls deps n. By an adissible index s s,...,s n Z n, we ean an index with positive entries and s 2 the latter condition is necessary for the convergence of the series in and 2. We also assign these characteristics to an MZV ζs or an MZSV ζ s theselves, just speaking about the weight and depth of the ultiple zeta star value, and calling it adissible if s is an adissible index. Apart fro a siplicity of several identities in ters of MZSVs a good exaple is the cyclic su forula cf. [5] versus [9], these zeta values appear very naturally in the diophantine probles for the values of the classical Rieann zeta function ζs, especially due to the identities 3a 3b ζ {2} n, 2ζ2n +, ζ {2} n 2 2 2n ζ2n, where {2} n denotes the n-tuple 2, 2,...,2; the interested reader is referred to the papers by Zlobin [20 22] we discuss these aspects at the end of Section. Identity 3a also coes as a special case of the cyclic su forula [5] while 3b follows fro a generating series arguent [, 22]. The starting goal of our project was to find a general for of identities 3a and 3b, and on its way, we succeeded at least in generalizing 3a see the two-one forula accopanied with Theores and 2 below. Another result Theore 3 appearing as an auxiliary identity in They also appear as non-strict MZVs in [4], the nae which is owed to the anonyous referee of that paper. Note also that the notation ζ instead of ζ was used in the Russian literature [7, 22, 23].

3 Zeta stars 327 our deduction of Theore 2, which is new and interesting by itself, is a weighted version of Euler s forula [5] 4 n ζl, n l ζn, n 3 l2 the su forula of depth 2 in the odern terinology. In Section, we provide forulae for expressing the values 2 in ters of and vice versa, and state our ain results Theores to 3. Section 2 is devoted to the proof of Theore, while Theores 2 and 3 are proved in Section 3.. Background and forulae for ultiple zeta star values Let us first consider a siple recipe of passing fro MZSVs to MZVs and vice versa. Proposition. For any adissible index s s,s 2,...,s n, we have the dual relations 5 ζ s p ζp and ζs p σp ζ p, where p runs through all indices of the for s s 2 s n with being either the sybol, or the sign +, and the exponent σp denotes the nuber of signs + in p. The total nuber of such indices p is 2 n. This stateent is alost obvious at least for the first identity in 5; the second one follows by the Möbius inversion. It also coes as a special case of Propositions 2 and 3 in [7] for generalized polylogariths 6 Li s z a > >a n z a a s asn n and Le s z a a n z a a s asn n at the point z. The following identity was discovered experientally when we searched for a generalization of 3a which is the particular case l of our finding. It is a weighted analogue of the expressions on the right-hand sides in 5. Two-one forula. For k 0,, 2,..., denote μ 2k+ {2} k,. Then for any adissible index s s,s 2,...,s l with odd entries s,...,s l, the

4 328 Yasuo Ohno and Wadi Zudilin following identities are valid: 7a 7b ζ μ s,μ s2,...,μ sl p p σp 2 lσp ζ p 2 lσp ζp, where, as before, p runs through all indices of the for s s 2 s l with being either the sybol, or the sign +, and the exponent σp denotes the nuber of signs + in p. Proof of the equality of the right-hand sides in 7a and 7b. By Proposition for the right-hand side in 7a, we have #{ +} 2 l#{ +} ζ s s 2 s l {, or +} {, or +} { + or } l#{ } 2 #{ } + ζs s 2 s l, which in the notation r #{ } + turns out to be l l n rn ln l n l n lr 2 r l r l#{ +}n l n ln 2 n + l n 0 ζs s 2 s l #{ +} ln l 2 n n {, or +} #{ +} ln ζs s 2 s l 2 l#{ +} ζs s 2 s l, ζs s 2 s l which is exactly the right-hand side of 7b. In spite of a nicely siple but soehow unusual for of the two-one forula, we cannot yet prove it in the full generality. The following two particular cases l 2, and s 3, s 2 s n2 with n l +2 3 arbitrary as well as our experiental results for cases not included in the theores below strongly support the validity of identities 7a, 7b.

5 Zeta stars 329 Theore. For any n and i n, 8 ζ 2,...,2,, 2,...,2, 4ζ 2i +, 2n 2i +2ζ2n +2. }{{}}{{} i ni Theore 2. For any n 3, 9 ζ 2,,..., 2 n2#{ +} ζ3 }{{}} {{} n2 {, or +} n3 2 ni ζ3 + e, +e 2, +e 3,...,+e ni, n i2 e +e 2+ +e nii2 where all e j are non-negative integers. We deduce Theore 2 fro the following weighted analogue of Euler s forula 4. Theore 3 Weighted su forula. For any n 3, 0 n 2 l ζl, n l n +ζn. l2 Before proceeding with our proofs of Theores to 3, let us ake soe coents on the two-one forula. The forula ζ {2, {} } n, +ζ +n + for any positive integers, n is known two different proofs are given in [5, 22]. If it is nothing but forula 3a, while if 2 then its lefthand side equals ζ {μ 3, {μ } 2 } n,μ. This together with the two-one forula ean that the corresponding right-hand side in 7a equivalently, in 7b is expected to have a closed-for evaluation by eans of the single zeta value +ζ +n +, where the integers 2 and n are arbitrary.

6 330 Yasuo Ohno and Wadi Zudilin Using the integral representation of MZSVs, ζ dx dx s+ +s s l l [0,] s + +s l i x x s+ +s i valid for any adissible index s s,...,s l, we can write the right-hand side of 7a as follows: a 2 [0,] s + +s l l i + x x s+ +s i l i x x s+ +s i dx dx s+ +s l. The change of variable y j x x j for j,...,s + + s l gives the integral b 2 >y > >y s + +s l >0 i l s+ +s i s + +s l js + +s l+ js + +s i+ dy j y j dy j dy s+ +s l, y j y s+ +s l + y s + +s i dy s+ +s i y s+ +s i y s+ +s i where the epty su s + + s i for i is interpreted as 0. Therefore, any of the two integrals in a, b ay replace the right-hand sides of 7a or 7b. The case l of the two-one forula is identity 3a. The case l 2 Theore reads as ζ {2} s,, {2} s2, 2ζ2s +2s ζ2s +, 2s 2 +. In particular, the latter identity iplies ζ {2} s,, {2} s2, + ζ {2} s2,, {2} s, 4ζ2s +2s ζ2s +, 2s 2 ++4ζ2s 2 +, 2s + 4ζ2s +ζ2s 2 +ζ {2} s, ζ {2} s2, whenever s and s 2. On the right-hand side of 7a and 7b, we have MZSVs and MZVs of length at ost l, while the left-hand side involves a single zeta star attached to an index with entries 2 and only and the nuber of s is equal to l; the latter circustance is the reason of dubbing the forula as the two-one forula.

7 Zeta stars 33 We stress that neither the two-one forula nor its special cases treated in Theores and 2 is specializations of identities for polylogariths 6 as ay be seen fro the fact Le 2, z 2Le 3 z for z cf. forula 3a with n. Zlobin s theore [2, Theore 3] iplies that under certain natural restrictions on positive integer paraeters b i >a i, i,...,, and c j, j,...,l, and with a choice of integers 2 r <r 2 < <r l such that r j+ r j is or 2 for all j,...,l, the ultiple integral 2 [0,] j xai i x i biai l j zx dx dx x 2 x rj cj is a Q[z ]-linear cobination of generalized polylogariths Le s z with indices s of weight at ost and length at ost l whose entries are only 2 and. In several cases, the ethod in [2] allows to control the nuber of s in these indices for instance, the case when only the zeta stars fro 3a occur after specialization z is treated in [2] in details. The integrals 2 and forulae 7a and 7b ay therefore becoe useful in diophantine study of ultiple zeta star values of depth higher than. 2. Depth 2 case of the two-one proble: Proof of Theore For a c>0, we define the haronic su 3 Ha, c c j j a and interpret H,c and Ha, 0 as zero. a j Lea. If B C, we have 4 A a B C c D a 2 c C c D A a B HB,c HA +,c c 2 Ha, C Ha, D a 2 + δ B,C 2 B 3.

8 332 Yasuo Ohno and Wadi Zudilin Proof. It follows that whenever a C, and A a B a c C c D c a a c a Ha, C Ha, D a c HB,c HA +,c+δ c,b c whenever c B. Furtherore, for a c, the following partial fraction decoposition is valid: 5 a 2 c a c a c 2 a c a 2. Thus, under the condition B C, weget A a B C c D a 2 c A a B C c D a c C c D A a B a c a c 2 a c HB,c HA +,c c 2 a 2 + δ B,C B 3 Ha, C Ha, D a 2 + δ B,C 2 B 3, which is the desired stateent. Reark. The proof of the cyclic su theore in [5] exploits the general fors of identities 5 and 4 which are l a +l c l a c a c a c a

9 Zeta stars 333 and A a B C c D l a +l c l C c D A a B HB,c HA +,c c Ha, C Ha, D a + δ B,C B +, respectively, although the function 3 is not used there in an explicit for it was introduced later by D. Zagier in his unpublished note on the proof in [5]. We are wondering if the two-one forula ay be generalized to soe kind of ultiple cyclic forula. Lea 2. For any i and j 0 we have 6 ζ 2i +, 2,...,2, }{{} j a 0 a a j Ha 0 +,a j a 2i+ 0 a 2 a2 j Proof. If j, we apply Lea with a a and c a j+ 7 ζ 2i +, 2,...,2, }{{} j a 0 a 2 a j+ a 0 a a j a 0 a a j a 0 a a j+ +2ζ 2i +, 2j +. a0 2i+ a 2 a2 j a j+ Ha 2,a j+ Ha 0 +,a j+ a 2i+ 0 a 2 2 a2 j a2 j+ Ha,a j a 2i+ 0 a 2 a2 2 a2 j Ha 0 +,a j a 2i+ 0 a 2 a2 j +2 a 0 a a 2i+ 0 a 2j+ +2ζ 2i +, 2j +. Although the application of 4 is only possible if j, it is easy to see that the resulting forula 6 trivially holds for j 0, since a 0 Ha 0 +,a 0 a 2i+ 0 in this case. a 0>b 0 a 2i+ 0 a 0 b a 0 a a 2i+ 0 a ζ 2i +, We will also need the following forula whose proof is just changing the condition a 0 a by a 0 >a in 7.

10 334 Yasuo Ohno and Wadi Zudilin Lea 3. For any i and j 0 we have a 0>a a j+ Proof. We proceed as in 7 a 0>a a j+ a 0 a j+ a 0 a 2i 0 a 2 2ζ2i +, 2j +. a2 j+ a 0>a a j a 0>a a j+ a 2i+ 0 a 2 a2 j a j+ Ha 0,a j a 2i+ 0 a 2 a2 j +2 a 0>a a 2i+ 0 a 2j+ a 2i+ 0 a 2 a2 j a +2ζ2i +, 2j + ; 0 a j+ again the forula reains valid for j 0. Then, we use the identity a 2i+ 0 a j+ + 0 a 0 a j+ a 2i 0 a j+a 0 a j+ a 0 a j+ a 0 a 2i 0 a 2 j+ a 2i+ to conclude with the desired clai. Lea 4. Given n, for any i n, the following identity holds: 8 ζ 2,...,2,, 2,...,2, 4ζ 2i +, 2n 2i + }{{}}{{} i ni Ha 0,a i Ha 0 +,a n + a 2 0 a2 n a 2 n a0 2i+ a 2. i+ a2 n a 0 a n a 0 a i+ a n Proof. We use Lea with a a 0 and c a i ζ 2,...,2,, 2,...,2, }{{}}{{} i ni a 0 a i a i+ a n+ a 2 a2 i a ia 2 i+ a2 na n+

11 a a i a i a i+ a n+ +2 a 0 a a i a i+ a n+ a 0 a i+ a n+ a 0 a a i a i+ a n+ Zeta stars 335 Ha,a i a 2 a2 i a2 i a2 i+ a2 na n+ Ha 0,a i Ha 0,a i+ a 2 0 a2 a2 i a2 i+ a2 na n+ a 2i+ 0 a 2 i+ a2 na n+ Ha 0,a i+ a 2 0 a2 a2 i a2 i+ a2 na n+ +2ζ 2i +, 2,...,2, }{{} ni Ha 0,a i a 2 a 0 a n 0 a2 n a +2ζ 2i +, 2,...,2,, n }{{} ni and for the latter zeta star ter, we apply Lea 2. Lea 5. We have ζ 2,...,2,, 4ζ 2n +, 2ζ2n +2. }{{} n To prove Lea 5, we will require an additional identity. Lea 6. For any positive integers l and satisfying l >, the following identity is valid: 9 b Hl, b l b a a al Proof. For the right-hand side in 9, we have Ha +,+. 20 al a Ha +,+ a al a a a d d0

12 336 Yasuo Ohno and Wadi Zudilin a a d aa d al d0 a 2 + d a a d al a 2 b b b b d d0 d a d a l b 2 + d d d al d0 a d a l b 2 + d d0 l b 2 + b b al a a d cd+ cb+ l b 2 + b b cb+ l c d d l c b b c while the right-hand side can be written as follows: b b c l c, d c l c l c 2 b Hl, b l b b b b b b c l bl c l b 2 + b l b 2 + d l b 2 + d b c d c l d b c l b l c d c d c l d d d c c d c l c d c dc+ l c d c

13 Zeta stars 337 b b b l b 2 + d l d d b l b 2 + b b db+ l b 2 + b b cb+ b c l d b c b c l c b b b c l c. b l c Coparing the right-hand sides of the resulting expressions in 20 and 2 and using cb+ b c c b c c cb+ cb+ b, c we derive the required identity 9. Proof of Lea 5. We are first interested in the finite ultiple su 22 F n a, b a 0 a n a 0 a, a n>b a a n a a, a n>b a 0 a n a 0 a, a n>b a 0 a n a 0 a, a n>b a 2 0 a2 n a n Ha,a n Ha +,a n a 2 a2 n a2 n Ha 0,a n Ha 0,b a 2 0 a2 a2 n Ha 0,b Ha +,a n a 2 0 a2 a2 n +2 a a 0>b +2 a a 0>b a 2n+ 0 a 2n+ 0. Multiplying both expressions for the su F n a, b in 22 by /a b /a b +, notifying that a b a b + a a 0 a 0 b

14 338 Yasuo Ohno and Wadi Zudilin and a n b a b a b + and suing over all a and b, a>b, we obtain 23 a 0 a n>b a a n + a, Ha 0,b a 0 ba 2 0 a2 n a n a 0 ba 2 a 0 a n>b 0 a2 a2 n a a n + Ha +,an a a 2 a a 0 a n> 0 a2 a2 n +2 a 0 ba0 2n+. a 0>b Applying Lea 6 with l a 0 and a n, we derive fro 23 that a 0 ba 2 0 a2 n a 2 n a 0 ba0 2n+. a 0 a n>b a 0>b Finally, fro Lea 4 in the case i n, we obtain ζ 2,...,2,, 4ζ 2n +, +2 }{{} a n a 0 ba 2n+ 0>b 0 2 a 0 ba0 2n+ 4ζ 2n +, 2ζ2n +2, a 0>b 0 which is the desired identity. Lea 7. For i<n, we have 24 a 0 a n 2 Ha 0,a i Ha 0,a i+ a 2 0 a2 n a n Ha 0,a n a 2i+ 0 a 2 i+ a2 n Ha 0,a n Ha 0,a n a 2 0 a2 i a ia 2 i+ a2 n a 2 0 a2 i a i+a 2. i+2 a2 n a 0>a i+ a n a 0 a n

15 Zeta stars 339 Proof. We use the identity in 22 for F i a, a i+ that is, for n replaced by i and b replaced by a i+. Multiplying it by /a 2 i+ a2 n and suing over a i+,...,a n+ satisfying a i+ a n a n+ result in the following identity: F i a, a n+ a 0 a i a i+ a n a 0 a, a n a n+ a a i a i+ a n a a, a n a n+ +2 a a i+ a n a a, a n a n+ a 2 0 a2 i a ia 2 i+ a2 n Ha,a i+ Ha +,a i a 2 a2 n a 2i+ a 2. i+ a2 n As before, we ultiply both sides by /a a n+ /a a n+ + and su over all a a n+ to obtain Ha 0,a n a 0 a n a 2 0 a2 i a ia 2 i+ a2 n a 0 a n a n+ a 0 a n+ a a n a n+ a a n+ a a a n a a n +2 a a i+ a n+ a a n+ a 0 a n+ a 2 0 a2 i a ia 2 i+ a2 n Ha,a i+ a a n+ a 2 a2 n a2 n Ha +,ai a a n a a 2 a2 n a2 n a a n+ a 2i+ a 2 i+ a2 n we set a 0 a in the second su a a n a n+ a a n+ Ha,a i+ a a n+ a 2 a2 n a2 n

16 340 Yasuo Ohno and Wadi Zudilin a 0 a a n a 0 a n +2 a a i+ a n a 0 a n a 0 a n a 0 a a n a 0 a n +2 a 0 a n a 0 Ha,a n a 2i+ a 2 i+ a2 n Ha 0,a i a 0 a n a 2 0 a2 n a 0 a i+ a n a 0 a n a 0 Ha 0,a n a0 2i+ a 2. i+ a2 n Ha0,a i +/a 0 a 2 a2 n a2 n Ha0,a i +/a 0 a 2 a2 n a2 n Using the partial fraction decoposition 5 with x a 0 reduces the latter expression to and y a n 25 a 0 a n +2 Ha 0,a n a 2 0 a2 i a ia 2 i+ a2 n a 0 a a n a 0 a n a 0 a i+ a n a 0 a n a 0 a a n a 0 a n a 0 a 2n+2 0 a 0 a n a 0 Ha 0,a n a 2i+ 0 a 2 i+ a2 n Ha 0,a i a 2 0 a2 n a n +2 + a 0 a 0 a n a 0 a n a 0 a n a 0 a 2 + a2 n a 0 a i+ a n /a 0 a 2 a2 n a2 n Ha 0,a 0 a 2n+ 0 Ha 0,a i a 2 0 a2 n a n a 0 a a n Ha 0,a n a0 2i+ a 2. i+ a2 n a 2 0 a2 a2 n

17 Zeta stars 34 Subtracting now the expression in 25 fro the one corresponding to the i replaced by i +, we finally obtain Ha 0,a n Ha 0,a n a 2 a 0 a n 0 a2 i a i+a 2 i+2 a2 n a 2 0 a2 i a ia 2 i+ a2 n Ha 0,a i Ha 0,a i+ a 2 a 0 a n 0 a2 n a n Ha 0,a n +2 a 2i+3 a 0 a i+2 a n 0 a 2 i+2 a2 n Ha 0,a n 2 a 2i+ 0 a 2, i+ a2 n a 0 a i+ a n which gives us the required forula 24. Lea 8. For 0 i<n, we have Ha 0,a n Ha 0,a n a 2 0 a2 i a ia 2 i+ a2 n a 2 0 a2 i a i+a 2 i+2 a2 n a 0 a n ζ 2,...,2,, 2,...,2, 2ζ2n +2. }{{}}{{} i+ ni Proof. We apply Lea, this tie with a a 0 and c a n+ ζ 2,...,2,, 2,...,2, }{{}}{{} a 2 i+ ni a 0 a n+ 0 a2 i a i+a 2 i+2 a2 na n+ Ha,a n+ a 2 a2 i a i+a 2 i+2 a2 n+ a a n+ a 0 a n a 0 a n a 0 a n Ha 0,a n a 2 0 a2 i a i+a 2 +2ζ2n +2 i+2 a2 n Ha 0,a n a 2 0 a2 i a ia 2 i+ a2 n Ha 0,a n a 2 0 a2 i a i+a 2 +2ζ2n +2, i+2 a2 n fro which the desired identity follows. We are now ready to prove Theore.

18 342 Yasuo Ohno and Wadi Zudilin Proof of Theore. We will use the descending induction on i n, n,...,. In the case i n induction base, the identity of the theore is already shown in Lea 5. Therefore, we assue that i<n and that identity 8 is proved with i replaced by i +, that is, 26 ζ 2,...,2,, 2,...,2, 4ζ 2i +3, 2n 2i 2ζ2n +2. }{{}}{{} i+ ni We substitute expressions Ha 0,a n a 2i+ a 0>a i+ a n 0 a 2 i+ a2 n Ha 0 +,a n a 2i+ a 0>a i+ a n 0 a 2 i+ a2 n + a 0 a n a 0 a0 2i+ a 2 i+ a2 n a 0>a i+ a n a 0>a i+ a n Ha 0 +,a n a 2i+ 0 a 2 +2ζ2i +3, 2n 2i i+ a2 n followed fro Lea 3 and Ha 0,a n Ha 0,a n a 2 0 a2 i a ia 2 i+ a2 n a 2 0 a2 i a i+a 2 i+2 a2 n a 0 a n 4ζ 2i +3, 2n 2i 4ζ2n +24ζ2i +3, 2n 2i followed fro Lea 8 and 26 into the identity of Lea 7 to get Ha 0,a i Ha 0,a i+ Ha 0 +,a n a 2 a 0 a n 0 a2 n a 2 n a 2i+ a 0>a i+ a n 0 a 2 i+ a2 n Ha 0 +,a n Ha 0 +,a n 2 a 2i+ 0 a 2 2 i+ a2 n a0 2i+3 a 2. i+2 a2 n a 0 a i+ a n The last identity ay be written as Ha 0,a i a 2 0 a2 n a 2 n a 0 a n a 0 a n Ha 0,a i+ a 2 0 a2 n a n a 0 a i+ a n 2 a 0 a i+2 a n a 0 a i+2 a n Ha 0 +,a n a 2i+ 0 a 2 i+ a2 n Ha 0 +,a n a0 2i+3 a 2, i+2 a2 n

19 Zeta stars 343 where the right-hand side equals 2ζ2n + 2 by Lea 4 applied to i + instead of i and 26, so does the left-hand side: 27 a 0 a n Ha 0,a i a 2 0 a2 n a n 2 a 0 a i+ a n Ha 0 +,a n a 2i+ 0 a 2 i+ a2 n 2ζ2n +2. Finally, fro 8 and 27 we obtain identity 8 for the given i, copleting the proof of Theore. 3. Weighted su forula: Proof of Theores 2 and 3 We first prove the weighted su forula. Proof of Theore 3. Recall two standard ways of coputing the product of two zeta values [23]: the shuffle product coing fro the representation of MZVs by iterated integrals [8] 28a n l ζζn l2 l + ζl, n l n and the haronic or stuffle product originated by the series representation [8] 28b ζζn ζn+ζ, n +ζn, ; here > and n>+ are arbitrary integers. 2 Suing up the righthand sides in 28a and 28b over 2, 3,...,n 2, we get n l2 which reduces to n 2 l2 n2 l 2 in{l,n2} 2 l + ζl, n l n n2 n3ζn+ ζ, n +ζn,, l 2 ζl, n l n 3ζn+2 n2 2 ζ, n. 2 A general for of these relations, known as double-shuffle relations, is discussed in [6, 0].

20 344 Yasuo Ohno and Wadi Zudilin Adding 2ζn, to the both sides of the latter equality, we obtain n 2 l2 l 2 l ζl, n l n 3ζn+2 n 2 ζ, n and using the binoial theore for the left-hand side results in 2 l ζl, n l n3ζn+2 n 2 l2 n 2 ζ, n. It reains to apply Euler s forula 4 to the both sides, to arrive at the desired forula 0. Reark 2. After reading a draft version of this paper Kentaro Ihara and Masanobu Kaneko pointed out that Theore 3 can also be proved by using generating functions in [6, 0] for instance, specializing X Y equality 3.4 in [6], and, in fact, there are ore weighted versions of Euler s relation 4 but they do not have such a siple for as in 0. Proof of Theore 2. For i 2,...,n, the su ζ3 + e, +e 2, +e 3,...,+e ni can be written as e +e 2+ +e nii2 e +e 2i2 i ζn i ++e, +e 2 ζn l, l by the identity in [3]. Therefore, 2 n2 ζ3,,...,+2 n3 ζ3 + e }{{}, +e 2,...,+e n3 + n3 e +e 2+ +e n3 +2 ni ζ3 + e, +e 2,...,+e ni + +2ζn n i2 n2 e +e 2+ +e nii2 2 ni i ζn l, l l 2 nl 2ζn l, l. l n2 l nl i2 l 2 i ζn l, l

21 Zeta stars 345 Applying Euler s forula 4, the latter su becoes n2 2 nl ζn l, l 2ζn, l which turns out to be n ζn by Theore 3. Finally, we use the forula n ζn ζ 2,,,...,, }{{} n2 which is a special case of the su forula due to Granville [7] and Zagier [9]. This iplies 9 and copletes our proof of Theore 2. As one can see fro the above proofs of Theores 2 and 3, these theores are equivalent odulo the algebraic relations fro [3] which also contain the su forula as a particular case. Acknowledgent We would like to thank Kentaro Ihara, Masanobu Kaneko, Jun-ichi Okuda and Don Zagier for useful coents. We are also thankful to Tatsushi Tanaka for providing us with a progra of coputing ultiple zeta star values which was used in our experiental verification of the forulae. We are indebted to the two anonyous referees of the journal for their helpful rearks. The work was done when both the authors visited in the Max Planck Institute for Matheatics Bonn. We thank the staff of the institute for the wonderful working conditions we experienced there. The work of Yasuo Ohno was supported by a special prograe of the Kinki University Osaka. The work of Wadi Zudilin was supported by a fellowship of the Max Planck Institute for Matheatics Bonn. References [] T. Aoki and Y. Ohno, Su relations for ultiple zeta values and connection forulas for the Gauss hypergeoetric functions, Publ. Res. Inst. Math. Sci , [2] D.J. Broadhurst and D. Kreier, Association of ultiple zeta values with positive knots via Feynan diagras up to 9 loops, Phys. Lett. B ,

22 346 Yasuo Ohno and Wadi Zudilin [3] P. Deligne and A.B. Goncharov, Groupes fondaentaux otiviques de Tate ixte, Ann. Sci. École Nor. Sup , 56. [4] V.G. Drinfel d, On quasitriangular quasi-hopf algebras and on a group that is closely connected with GalQ/Q, Algebra i Analiz 990, 49 8; Leningrad Math. J. 2 99, English transl.. [5] L. Euler, Meditationes circa singulare serieru genus, Novi Co. Acad. Sci. Petropol , 40 86; Opera Onia, Ser. I, vol. 5, Teubner, Berlin, 927, Reprinted. [6] H. Gangl, M. Kaneko and D. Zagier, Double zeta values and odular fors, Autoorphic fors and zeta functions, Proceedings of the Conference in Meory of Tsuneo Arakawa, World Sci. Publ., Hackensack, NJ, 2006, [7] A. Granville, A decoposition of Rieann s zeta-function, Analytic nuber theory Kyoto, 996, London Matheatical Society Lecture Note Series, 247, Cabridge University Press, Cabridge, 997, [8] M.E. Hoffan, Multiple haronic series, Pacific J. Math , [9] M.E. Hoffan and Y. Ohno, Relations of ultiple zeta values and their algebraic expression, J. Algebra , [0] K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for ultiple zeta values, Copos. Math , [] C. Kassel, Quantu groups, Graduate Texts in Matheatics, 55, Springer-Verlag, New York, 995. [2] D. Kreier, Knots and Feynan diagras, Cabridge Lecture Notes in Physics 3, Cabridge University Press, Cabridge, [3] Y. Ohno,A generalization of the duality and su forulas on the ultiple zeta values, J. Nuber Theory , [4] Y. Ohno and J. Okuda, On the su forula for the q-analogue of nonstrict ultiple zeta values, Proc. Aer. Math. Soc , [5] Y. Ohno and N. Wakabayashi, Cyclic su of ultiple zeta values, Acta Arith ,

23 Zeta stars 347 [6] T. Terasoa, Mixed Tate otives and ultiple zeta values, Invent. Math , [7] E.A. Ulanskii, Identities for generalized polylogariths, Mat. Zaetki , ; Math. Notes , English transl.. [8] D. Zagier, Values of zeta functions and their applications, First European Congress of Matheatics, vol. II Paris, 992, Progress in Matheatics, 20, Birkhäuser, Basel, 994, [9] D. Zagier, Multiple zeta values, Unpublished anuscript, Bonn, 995. [20] S.A. Zlobin, On soe integral identities, Uspekhi Mat. Nauk 2002, 53 54; Russian Math. Surveys , English transl.. [2] S.A. Zlobin, Expansion of ultiple integrals in linear fors, Mat. Zaetki , ; Math. Notes , English transl.. [22] S.A. Zlobin, Generating functions for the values of a ultiple zeta function, Vestnik Moskov. Univ. Ser. I Mat. Mekh. no , 55 59; Moscow Univ. Math. Bull , English transl.. [23] W. Zudilin, Algebraic relations for ultiple zeta values, Uspekhi Mat. Nauk 2003, 3 32; Russian Math. Surveys , 29 English transl.. Departent of Matheatics Kinki University Higashi-Osaka Osaka Japan E-ail address: ohno@ath.kindai.ac.jp Departent of Mechanics and Matheatics Moscow Loonosov State University Vorobiovy Gory, GSP- 999 Moscow Russia Steklov Matheatical Institute Russian Acadey of Sciences Gubkina str Moscow Russia E-ail address: wadi@i.ras.ru Received Noveber, 2007

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