MULTIPLE ZETA VALUES 45

Transcription

1 MULTIPLE ZETA VALUES The two-one conjectural formula In the introductory section the following alternative version of the multiple zeta values with non-strict inequalities was mentioned see.7: ζ s ζ s, s 2,..., s l : n n 2 n l n s n s 2 2 n s l Exercise.2 gives a simple recipe to pass from one model to the other. Relation.8 is an example of simple relations for the multiple zeta star values; its companion is ζ {2} k 2 2k ζ2k 2 n n n 2k. This expression can be compared with the one for ζ{2} k given in 5.4 and 2.. The starting goal of our joint project with Y. Ohno was finding a general form of the two families of identities for the MZSVs. On this way, we only succeeded in generalising.8 but conjecturally. The particular cases of our conjecture which we dubbed as the two-one formula were established by ourselves; there are some recent publications with some other particular instances proven. One of lucky accidents of our proofs is the weighted version 2.8 of Euler s original formula 2.7 the sum formula of depth 2 in the modern terminology. Conjecture 7 Two-one formula. For k 0,, 2,..., denote µ 2k+ {2} k,. Then for any admissible index s s, s 2,..., s l with odd entries s,..., s l, the following identities are valid: l. ζ µ s, µ s2,..., µ sl p p σp 2 l σp ζ p 4. 2 l σp ζp, 4.2 where, as in Exercise.2, p runs through all indices of the form s s 2 s l with being either the symbol, or the sign +, and the exponent σp denotes the number of signs + in p. Proof of the equality of the right-hand sides in 4. and 4.2. By Exercise.2 for the right-hand side in 4. we have {, or +} #{ +} 2 l #{ +} ζ s s 2 s l {, or +} { + or } l #{ } #{ }+ ζs s 2 s l

2 46 JONATHAN M. BORWEIN AND WADIM ZUDILIN which in the notation r #{ } + turns out to be l l n rn l n l n m0 l 2 n n {, or +} l n l r 2 r l r l n l n m #{ +}l n l #{ +}n l n m 2 n+m ζs s 2 s l 2 l #{ +} ζs s 2 s l, and this is exactly the right-hand side of 4.2. ζs s 2 s l #{ +}l n ζs s 2 s l On the right-hand side of 4. and 4.2 we have MZSVs and MZVs of length at most l, while the left-hand side involves a single zeta star attached to an index with entries 2 and only and the number of s is equal to l; the latter circumstance is the reason of dubbing the formula as the two-one formula. We stress that neither the two-one formula nor its special cases treated in Theorems 4. and 4.2 below are specializations of identities for polylogarithms 4.. In spite of a nicely simple but somehow unusual form of the two-one formula we cannot yet prove it in the full generality. Besides the case l given in.8, the following two particular cases l 2, and s 3, s 2 s n 2 with n l arbitrary as well as our experimental results for cases not included in the theorems below strongly support the validity of identities 4., 4.2. Theorem 4.. For any n and i n, ζ 2,..., 2,, 2,..., 2, 4ζ 2i +, 2n i + ζ2n i n i Theorem 4.2. For any n 3, ζ 2,,..., 2 n 2 #{ +} ζ3 } {{ } n 2 {, or +} n 3 n 2 n i ζ3 + e, + e 2, + e 3,..., + e n i, i2 e +e 2 + +e n i i 2 where all e j are non-negative integers. 4.4 Before giving some details of proofs of the theorems, let us make some comments on the two-one formula. The formula ζ {2, {} m } n, m + ζm + n +

3 MULTIPLE ZETA VALUES 47 for any positive integers m, n is known two different proofs are given by Zlobin and Ohno Wakabayashi. If m it is nothing but formula.8, while if m 2 then its left-hand side equals ζ {µ 3, {µ } m 2 } n, µ. This together with the twoone formula mean that the corresponding right-hand side in 4. equivalently, in 4.2 is expected to have a closed-form evaluation by means of the single zeta value m + ζm + n +, where the integers m 2 and n are arbitrary. Using the integral representation of MZSVs, ζ s [0,] s + +s l s + +s l l i t t s + +s i cf. 7.0 valid for any admissible index s s,..., s l, we can write the righthand side of 4. as follows: l i 2 + t t s + +s i l i t t s + +s i s + +s l. 4.5 [0,] s + +s l The change of variable u j t t j for j,..., s + + s l gives the integral l s + +s i du j 2 u j >u > >u s + +s l >0 i s + +s l js + +s l + js + +s i + + u s + +s i du s + +s i u s + +s i u s + +s i du j du s + +s l, 4.6 u j u s + +s l where the empty sum s + +s i for i is interpreted as 0. Therefore, any of the two integrals in 4.5, 4.6 may replace the right-hand sides of 4. or 4.2. The case l 2 Theorem 4. reads as ζ {2} s,, {2} s 2, 2ζ2s + 2s ζ2s +, 2s 2 +. In particular, the latter identity implies ζ {2} s,, {2} s 2, + ζ {2} s 2,, {2} s, 4ζ2s + 2s ζ2s +, 2s ζ2s 2 +, 2s + 4ζ2s + ζ2s 2 + ζ {2} s, ζ {2} s 2, whenever s and s 2. However, no further generalizations to cases l > 2 can be derived from Conjecture 7. The proof of Theorem 4. from the joint paper with Y. Ohno is an elaborate descending inductive argument on i. The following two exercises represent the summary of this proof given in eight lemmas. Exercise 4.. For a c > 0, define the harmonic sum c Ha, c a j j j a and interpret both H, c and Ha, 0 as zeroes.

4 48 JONATHAN M. BORWEIN AND WADIM ZUDILIN a If B C, we have a 2 c HB, c HA +, c c 2 Ha, C Ha, D a 2 + δ B,C 2 B 3, where δ B,C stands for Kronecker s delta. b For positive integers L and M satisfying L > M, the following identity is valid: M b HL, b l b c a al a M a Ha +, M +. Solution of part a. It follows that Ha, C Ha, D a c whenever a C, and a c a a c HB, c HA +, c + δ c,b c whenever c B. Furthermore, for a c the following partial fraction decomposition is valid: a 2 c a c a c 2 a c a. 2 Thus, under the condition B C, we get a 2 c a c a c 2 a c + δ a 2 B,C B 3 a c HB, c HA +, c c 2 which is the desired statement. l Ha, C Ha, D a 2 + δ B,C 2 B 3. Remark. The proof of the cyclic sum theorem Theorem 2.3 given by Ohno and Wakabayashi exploits the general forms of above partial-fraction identities: m a m+ l c l a c a c m a c a m

5 and m l a m+ l c l MULTIPLE ZETA VALUES 49 HB, c HA +, c c m Ha, C Ha, D a m + δ B,C m B m+, respectively, although the function Ha, c was not used there in an explicit form it was introduced later by Zagier in his unpublished note on the proof of Ohno Wakabayashi. It is an open question whether the two-one formula may be generalized further to some multiple cyclic level. Exercise 4.2. a For i and j 0, ζ 2i +, 2,..., 2, j a 0 >a a j+ a 0 a a j Ha 0 +, a j a 2i+ 0 a 2 a 2 j b For i and j 0, a 0 a j+ a 0 a 2i 0 a 2 a 2 j+ c Given n, for any i in the range i n, ζ 2,..., 2,, 2,..., 2, 4ζ 2i +, 2n i + i n i Ha 0, a i + a 2 0 a 2 n a n d We have a 0 a n a 0 a i+ a n ζ 2,..., 2,, 4ζ 2n +, ζ2n + 2. n a 0 a n + 2ζ 2i +, 2j +. 2ζ2i +, 2j +. Ha 0 +, a n a 2i+ 0 a 2 i e For i < n, Ha 0, a i Ha 0, a i+ Ha 0, a n 2 a 2 a 0 a n 0 a 2 n a n a 2i+ a 0 >a i+ a n 0 a 2 i+ Ha 0, a n Ha 0, a n a 2 0 a 2 i a ia 2 i+ a 2 0 a 2 i a i+a 2 i+2. f For 0 i < n, a 0 a n Ha 0, a n Ha 0, a n a 2 0 a 2 i a ia 2 i+ a 2 0 a 2 i a i+a 2 i+2 ζ 2,..., 2,, 2,..., 2, ζ2n + 2. i+ n i

6 50 JONATHAN M. BORWEIN AND WADIM ZUDILIN Proof of Theorem 4.. We will use the descending induction on i n, n,...,. In the case i n induction base the identity of the theorem is shown in Exercise 4.2 d. Therefore, we assume that i < n and that identity 4.3 is proved with i replaced by i +, that is, ζ 2,..., 2,, 2,..., 2, 4ζ 2i + 3, 2n i ζ2n i+ n i We substitute expressions a 0 >a i+ a n Ha 0, a n a 2i+ 0 a 2 i+ a 2i+ a 0 >a i+ a n + a 0 >a i+ a n a 0 >a i+ a n Ha 0 +, a n 0 a 2 i+ a 0 a n a 0 a 2i+ 0 a 2 i+ Ha 0 +, a n a 2i+ 0 a 2 i+ + 2ζ2i + 3, 2n i, followed from Exercise 4.2 b, and Ha 0, a n Ha 0, a n a 2 0 a 2 i a ia 2 i+ a 2 0 a 2 i a i+a 2 i+2 a 0 a n a 0 a n 4ζ 2i + 3, 2n i 4ζ2n + 2 4ζ2i + 3, 2n i, followed from Exercise 4.2 f and 4.8, into the identity of Exercise 4.2 e to get Ha 0, a i Ha 0, a i+ Ha 0 +, a n 2 a 2 0 a 2 n a n a 2i+ 0 a 2 i+ 2 a 0 a i+ a n Ha 0 +, a n a 2i+ 0 a 2 i+ The last identity may be written as Ha 0, a i a 2 0 a 2 n a n a 0 a n a 0 a n a 0 a n Ha 0, a i+ a 2 0 a 2 n a n a 0 a i+ a n a 0 >a i+ a n a 0 a i+2 a n a 0 a i+2 a n Ha 0 +, a n a 2i+ 0 a 2 i+ Ha 0 +, a n a 2i+3 0 a 2 i+2. Ha 0 +, a n a 2i+3 0 a 2 i+2, where the right-hand side equals 2ζ2n + 2 by Exercise 4.2 c applied to i + instead of i and 4.8, so does the left-hand side: Ha 0, a i Ha 0 +, a n a 2 0 a 2 n a n a 2i+ 0 a 2 i+ 2ζ2n a 0 a i+ a n

7 MULTIPLE ZETA VALUES 5 Finally, from 4.7 and 4.9 we obtain identity 4.3 for the given i, completing the proof of Theorem 4.. Proof of Theorem 4.2. For each i 2,..., n, the sum ζ3 + e, + e 2, + e 3,..., + e n i can be written as e +e 2 + +e n i i 2 e +e 2 i 2 i ζn i + + e, + e 2 ζn l, l by Theorem 8.4 Ohno s relations. Therefore, 2 n 2 ζ3,,..., + 2 n 3 ζ3 + e, + e 2,..., + e n 3 + n 3 e +e 2 + +e n n i ζ3 + e, + e 2,..., + e n i + + 2ζn n i2 2 n i i e +e 2 + +e n i i 2 n 2 ζn l, l l n 2 2 n l ζn l, l. l l n l i2 l 2 i ζn l, l Applying Euler s formula 2.7 and its weighted version 2.8, the latter sum becomes n 2 n 2 2 n l ζn l, l ζn l, l n ζn. l Finally, we use the formula l n ζn ζ 2,,,..., n 2 This im- which follows from Exercise.2 and the sum theorem Theorem 2.4. plies 4.4 and completes our proof of Theorem Reduction of double Euler sums Euler s original motivation to study double zeta sums was a possibility to reduce them to single zeta values. We have already discussed this problem in Section 3 for the standard multiple zeta values. Here we reproduce a result of Kentaro Ihara which addresses the alternating double Euler sums ζr, s;, σ n>m σ m n r m s,

8 52 JONATHAN M. BORWEIN AND WADIM ZUDILIN which in fact works for any choice of σ on the unit circle σ not just for σ {±}. First note the iterated integral representation ζr, s;, σ r r r+ r+s σ r+s t t r t r t r+ t r+s σt r+s >t > >t r+s >0 0 t r times t t s times σ σt, 5. which is given in the mnemonic form like in 4. in the second line. Theorem 5.. For k 2 and σ, ζk, ;, σ + ζk, ;, σ ζk ζ; σ + ζ; σ + kζk + k ζj; σζk j + ; σ. 5.2 j The particular case σ corresponds to the identity 2ζk, 2 where we use n>m m k n k m 2ζkζ + kζk + ζjζk j + ζk j 4 k ζk log 2 + kζk + k j j k ζjζk j +, 5.3 j2 { log 2 if k, k ζk if k >. Exercise 5.. Show that the limit of the right-hand side in 5.2 as σ, σ, exists and deduce the corresponding identity due to Euler k 2ζk, kζk + ζjζk j In order to prove Theorem 5. we use the shuffle and stuffle relations for the corresponding alternating double sums. j2 Lemma 5.. We have ζk ζ; σ + ζ; σ k ζj, k + j; σ, σ + ζk + j, j; σ, σ j + ζk, ;, σ + ζk, ;, σ.

9 Proof. The shuffle product of ζ; σ and ζk reads ζ; σζk MULTIPLE ZETA VALUES 53 0 k j σ σt 0 0 t j times t k times σ σt t t k j times t + 0 t σ t σt k times k ζj, k + j; σ, /σ + ζk, ;, σ. j It remains to add the equation obtained by replacing σ with σ. Lemma 5.2. The following identity is valid: k k ζj; σζk+ j; σ ζj, k+ j; σ, σ +ζk+ j, j; σ, σ +kζk+. j j Proof. By the shuffle product that is, term-by-term multiplication of the corresponding series, ζj; σζk + j; ε ζj, k + j; σ, ε + ζk + j, j; ε, σ + ζk + ; σε. Putting ε σ and summing for j from to k, the result follows. Proof of Theorem 5.. The identity follows by applying Lemmas 5. and 5.2.