Ohno-tye relation for finite multile zeta values Kojiro Oyama Abstract arxiv:1506.00833v3 [math.nt] 23 Se 2017 Ohno s relation is a well-known relation among multile zeta values. In this aer, we rove Ohno-tye relation for finite multile zeta values, which is conjectured by Kaneko. As a corollary, we give an alternative roof of the sum formula for finite multile zeta values, which was first roved by Saito and Wakabayashi. 1 Introduction 1.1 Finite multile zeta values The finite multile zeta value is defined by Zagier [5, 6]. Definition 1.1. The Q-algebra A is defind by / A : Z/Z Z/Z {a a Z/Z}/, :rime :rime where a b means a b for all but finitely many rimes. Definition 1.2. For any k k 1,..., k r Z 1 r, the finite multile zeta value is defined by 1 ζ A k ζ A k 1,..., k r : mod A. m k 1 1 m kr r >m 1 > >m r>0 For an index k k 1,..., k r Z 0 r, the integer k 1 + + k r is called the weight of k denoted by wtk and the integer r is called the deth of k denoted by dek. For k k 1,....k r and k k 1,..., k r, the symbol k + k reresents the comonentwise sum: k + k : k 1 + k 1,..., k r + k r. For any k k 1,..., k r, we define Hoffman s dual of k by By definition, we easily find k : 1,..., 1 + 1,..., 1 +1,..., 1 + 1,..., 1. }{{}}{{}}{{} k 1 k 2 k r dek + dek wtk + 1, k k. 2010 Mathematics Subject Classification Numbers: Primary 11M32. Secondary 05A19. Key words and hrases: finite multile zeta value, Ohno s relation, sum formula. 1
Examle 1.3. 2, 3, 1, 2 1, 1 + 1, 1, 1 + }{{}}{{}}{{} 1 + 1, 1 }{{} 2 3 1 2 1, 2, 1, 3, 1, 1, 2, 1, 3, 1 }{{} 1 + 1, 1 + }{{}}{{} 1 + 1, 1, 1 + }{{}}{{} 1 1 2 1 3 1 2, 3, 1, 2. Our main theorem is the following result conjectured by Kaneko [5, 6]: Theorem 1.4 Main Theorem. For any k 1,..., k r Z 1 r and n Z 0, we have ζ A k 1 + e 1,..., k r + e r ζ A + e 1,..., k s + e s, e 1 + +e rn e i 0 e 1 + +e sn e i 0 where k 1,..., k s k 1,..., k r is Hoffman s dual of k 1,..., k r Z 1 r. Remark 1.5. We can rove Ohno-tye relation for finite real multile zeta values, by exactly the same roof given in 2. 1.2 Algebraic setu and Notation Let H : Q x, y be the non-commutative olynomial algebra over Q in two indeterminates x and y, and H 1 : Q + Hy its subalgebra. Put z k : x k 1 y for a ositive integer k. Then H 1 is generated by z k k 1, 2, 3,.... For a word w H, the total degree of w is called the weight of w denoted by w and the degree in y is called the deth of w. The harmonic roduct on H 1 is the Q-bilinear ma : H 1 H 1 defined inductively by the following rules: i For any w H 1, w 1 1 w w. ii For any w 1, w 2 H 1 and m, n Z 1, z m w 1 z n w 2 z m w 1 z n w 2 + z n z m w 1 w 2 + z m+n w 1 w 2. The shuffle roduct X on H is the Q-bilinear ma X : H H defined inductively by the following rules: i For any w H, w X 1 1 X w w. ii For u 1, u 2 {x, y} and any w 1, w 2 H, u 1 w 1 X u 2 w 2 u 1 w 1 X u 2 w 2 + u 2 u 1 w 1 X w 2. It is known that these roducts are commutative and associative cf: [3, 8]. We define a Q-linear ma Z A : H 1 A by setting Z A 1 1, Z A z z kr ζ A k 1,..., k r. The following roosition gives the imortant roerties of Z A : Proosition 1.6 [3, 5, 6]. For any words w z z kr and w z k 1 z k s H 1, we have 1 Z A w w Z A wz A w, 2 Z A w X w 1 w Z A z kr z z k 1 z k s. 2
We define a Q-linear ma T : H 1 H 1 by T w : τw y for any w w y H 1, where τ is the automorhism of H given by τx y, τy x. The ma T reresents Hoffman s dual for words, namely, if Z A w ζ A k, then Z A T w ζ A k. 2 Proof of the main theorem 2.1 Preliminaries Proosition 2.1 [4, Corollary 3]. For any w H 1, we have where u is the automorhism of H given by 1 1 yu w 1 1 yu X uw, 1 u x x1 + yu 1, u y y + x1 + yu 1 yu, and u is a formal arameter. We define the symbols required to rove Theorem 1.4 by the following: Definition 2.2. For k k 1, k 2,..., k r Z 1 r, r Z 1, and n, l Z 0, we define a l k 1, k 2,..., k r : A k,l,n : e 1 + +e + +kr l e i 0 λ 1 + +λ rn λ j {0,1} j 1 1 xy e j 1 y k2 j 2 1 xy e k 1 +j 2 y kr a l k 1 1 + λ 1, k 2 1 + λ 2,..., k r 1 + λ r. j r1 xy e k 1 + +k r 1 +jr Lemma 2.3. For any k k 1, k 2,..., k r and n Z 1, we have 1 i Z A y k A k,l,i 0. 2 i0 k+ln i Proof. For w z z k2 z kr, we calculate both sides of 1. L.H.S. of 1 yu n z z k2 z kr n0 z1 n z z k2 z kr u n. n0 y, 3
Here, by the definition of u, we obtain u z z k2 z kr x1 + yu 1 x1 + yu 1 {y + x1 + yu 1 yu} }{{} k 1 1 x1 + yu 1 x1 + yu 1 {y + x1 + yu 1 yu} }{{} k 2 1 x1 + yu 1 x1 + yu 1 {y + x1 + yu 1 yu} }{{} k r 1 r 1 l A k,l,i u l+i. l0 i0 Thus we have R.H.S. of 1 yu k X k0 n0 i0 l0 r 1 l A k,l,i u l+i i0 r 1 l y k X A k,l,i u n+i. k+ln Comaring the cofficients of u n on both sides, we have 1 l y k X A k,l,i z1 n z z k2 z kr. 3 i0 k+ln i Alying Z A to both sides of 3 gives Z A L.H.S. of 3 Z A i0 i0 i0 k+ln i k+ln i k+ln i Z A R.H.S. of 3 Z A z n 1 z z k2 z kr Z A z n 1 Z A z z k2 z kr 1 l y k X A k,l,i 1 l Z A y k X A k,l,i 1 n i Z A y k A k,l,i, ζ A 1,..., 1ζ }{{} A k 1, k 2,..., k r 0. n The last equality is a consequence of [2, Equation15] which asserts that ζ A a,..., a 0 }{{} r 4
for any a, r Z 1. Multilying both sides by 1 n gives 1 i Z A y k A k,l,i 0. i0 k+ln i Lemma 2.4. For any k k 1, k 2,..., k r and n Z 1, we have i0 where s dek. 1i wtλi e Z 0 s+i wten i Proof. We first show the following equality: Z A y k a l k 1 1, k 2 1,..., k r 1 k+ln ζ A k + λ + e 0, e Z 0 s wten ζ A k + e, 4 where s dek. By the definition of a l k 1, k 2,..., k r, we easily find 1 kr 1 L.H.S. of 4 Z A y k xy e j 1 y xy e k 1 + +k r 1 r+1+jr y k+e 1 + +e + +kr rn k, e i 0 e 1 + +e + +kr r+1n e i 0 e 1 + +e + +kr r+1n e i 0 Z A y e 1 j 1 1 1 j 1 1 ζ A k + e, xy e j 1 +1 y j r1 kr 1 j r1 xy e k 1 + +k r 1 r+2+jr where the last equality holds by observing the following: The word which corresonds to the index k k 1,..., k r is and thus 1 j 1 1 T x k 1 1 yx k 2 1 y x kr 1 y y k 1 1 xy k 2 1 x y kr, x e j 1 y x k2 1 j 2 1 x e k 1 1+j 2 y x kr j r1 x e k 1 + +k r 1 r+1+jr y corresonds to k + e e e 1,..., e + +k r r+1. By taking Hoffman s dual, we see that the word which corresonds to k + e is 1 k2 1 kr T x e j 1 y x x e k 1 1+j 2 y x x e k 1 + +k r 1 r+1+jr y y e 1 j 1 1 1 j 1 1 xy e j 1 +1 j 2 1 y k2 1 j 2 1 xy e k 1 +j 2 y j r1 kr 1 j r1 xy e k 1 + +k r 1 r+2+jr y. y 5
Thus we find 4. Accordingly, we have L.H.S. of 2 i0 i0 i0 1 i 1i 1i k+ln i k+ln i wtλi Z A y k A k,l,i λ 1 + +λ ri λ j {0,1} e Z 0 s+i wten i 2.2 Proof of the main theorem Thorem 1.4 can be rewritten as follows: Z A y k a l k 1 1 + λ 1, k 2 1 + λ 2,..., k r 1 + λ r ζ A k + λ + e. Theorem 2.5 Theorem 1.4. For any k k 1, k 2,, k r and n Z 0, we have where s dek. e Z 0 r wten ζ A k + e e Z 0 s wte n ζ A k + e, Proof. Let us rove Theorem 1.4 by induction on n. First, it is trivial in the case n 0. Next, we assume that the equality holds for values smaller than n. Then the assumtion shows that ζ A k + λ + e ζ A k + λ + e wtλi e Z 0 s+i wten i e Z 0 r wtλi wte n i 1 µ 1 < <µ i r for i with 1 i min{n, r}. By Lemma 2.4, we obtain wtλ0 e Z 0 s wten ζ A k + λ + e i1 1i 1 wtλi e Z 0 r wte n e µ1,...,e µi 1 e Z 0 s+i wten i ζ A k + e ζ A k + λ + e. 6
Therefore we have ζ A k + e e Z 0 s wten e Z 0 s wtλ0 wten i1 i1 1i 1 1 i 1 Hence we must show the following identity: i1 1 i 1 1 µ 1 < <µ i r e Z 0 r wte n e µ1,...,e µi 1 ζ A k + λ + e wtλi e Z 0 s+i wten i 1 µ 1 < <µ i r ζ A k + λ + e e Z 0 r wte n e µ1,...,e µi 1 ζ A k + e e Z 0 r wten ζ A k + e. ζ A k + e. We fix the index e e 1,..., e r such that wte n, and let the number of non-zero comonents of the fixed index e be m 1 m min{n, r}. In the left-hand side, the number of aearances of the index e is Thus we have i1 m 1 1 i 1 m + 2 1 µ 1 < <µ i r m 3 e Z 0 r wte n e µ1,...,e µi 1 Therefore the identity holds for n. The roof is comlete. 3 Sum formula m m 1 i 1 1. i i1 ζ A k + e e Z 0 r wten The following identity was first roved by Saito and Wakabayashi [9]. k 1 + +k rk k j 1,k i 2 ζ A k + e. Corollary 3.1 Sum fomula [9, Theorem 1.4]. For k, r, i Z 1 with 1 i r k 1, we have k 1 k 1 ζ A k 1,..., k r 1 i 1 + 1 r B k mod. 5 i 1 r i k 7
Proof. Let k 1,..., 1, 2, 1,..., 1 and n k r 1. Then k }{{}}{{} i, r + 1 i and s 2. i 1 r i Aly Theorem 1.4 to this k to get: ζ A k + e e Z 0 r wtek r 1 e 1 + +e rk r 1 e j 0 k 1 + +k rk k j 1,k i 2 e Z 0 2 wte k r 1 e 1 +e 2 k r 1 e j 0 e 1 +e 2 k r 1 e 1 +e 2 k r 1 e j 0 ζ A 1 + e 1,..., 1 + e i 1, 2 + e i, 1 + e i+1,..., 1 + e r ζ A k 1,..., k r, ζ A k + e ζ A i + e1, r + 1 i + e 2 ζ A 1,..., 1, 2, 1,..., 1 }{{}}{{} e j 0 i 1+e 1 r i+e 2 r i+e 1 2 +1 k B k r i + e 2 + 1 k mod k i k 1 e B k mod by utting e r i + e 2 + 1 e k er i+1 k i r i k 1 e B k mod e k e0 e0 k 1 k 1 1 i 1 1 k+1 + 1 r B k mod. i 1 r i k Thus we have k 1 + +k rk k j 1,k i 2 ζ A k 1,..., k r 1 i 1 1 k+1 k 1 i 1 by [7, Theorem 4.5] k 1 + 1 r B k r i k mod. 6 If k is even, then the right-hand side of 5 is equal to the right-hand side of 6 because B k 0 whenever k + 3. If k is odd, then the right-hand side of the 5 is equal to the right-hand side of 6 because 1 k+1 1. Therefore the roof is comlete. Acknowledgment The author would like to exress his sincere gratitude to Professors Masanobu Kaneko and Shingo Saito for helful comments and advice. 8
References [1] M. E. Hoffman, Multile harmonic series, Pacific J. Math. 152 1992, 275 290. [2] M. E. Hoffman, Quasi-symmetric functions and mod multile harmonic sums, Kyushu J. Math. 69 2015, 345-366. [3] M. E. Hoffman, The algebra of multile harmonic series, J. Algebra 194 1997, 477 495. [4] K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for multile zeta values, Comositio Math. 142 2006, 307-338. [5] M. Kaneko, Finite multile zeta values. in Jaanese, RIMS Kôkyûroku Bessatsu, to aear. [6] M. Kaneko and D. Zagier, Finite multile zeta values, in rearation. [7] Kh. Hessami Pilehrood, T. Hessami Pilehrood, and R. Tauraso, New roerties of multile harmonic sums modulo and -analogues of Leshchiner s series, Trans. Amer. Math. Soc. 366 2014, 3131-3159. [8] C. Reutenauer, Free Lie Algebras, Oxford Science Publications, 1993. [9] S. Saito and N. Wakabayashi, Sum formula for finite multile zeta values, J. Math. Soc. Jaan 67 2015, 1069-1076. [10] S. Saito and N. Wakabayashi, Bowman-Bradley tye theorem for finite multile zeta values, Tohoku Math. J. 2 68 2016, 241-251. Kojiro Oyama E-mail: k-oyama@kyudai.j 9