aa r X i v : . [ m a t h . N T ] F e b A NOTE ON OHNO SUMS FOR MULTIPLE ZETA VALUES
HIDEKI MURAHARA
Abstract.
The Ohno relation is a well-known relation among multiple zeta values.Hirose, Onozuka, Sato, and the author investigated the sum related to the Ohno relationand presented two types of new relations and five conjectural formulas. This paper provesone of these formulas. Introduction
We call a sequence of positive integers “index” and an index whose last componentgreater than 1 “admissible index”. For an admissible index ( k , . . . , k r ), the multiple zetavalues (MZVs) are defined by ζ ( k , . . . , k r ) := X ≤ m < ··· Theorem 1.2 (Ohno relation; Ohno [3]) . For an admissible index k , we have O ( k ) = O ( k † ) . Mathematics Subject Classification. Primary 11M32; Secondary 05A19. Key words and phrases. Multiple zeta values, Ohno relation, Ohno sum. We denote by I the Q -linear space spanned by all indices. We define the Q -bilinearproduct e x : I × I → I by 1 e x k = k e x k , ( k , k ) e x ( l , l ) = ( k e x ( l , l ) , k ) + (( k , k ) e x l , l ) , where k , l ∈ I and k, l ∈ Z ≥ . For example, we have ( a ) e x ( b, c ) = ( a, b, c ) + ( b, a, c ) +( b, c, a ). For an admissible index k and an integer s ≥ 2, put F ( s ; k ) := O (( s ) e x k ) − O (( s ) e x k † ) . Then the following Ohno sum relation holds: Theorem 1.3 (Hirose–Murahara–Onozuka–Sato [1]) . For integers s, t ≥ , we have F ( s ; ( t + 1)) = F ( t ; ( s + 1)) . We denote by { a } l l -times repetition of a , e.g., ( { } ) = (2 , , Theorem 1.4 (Main theorem) . For integers s, t ≥ and l ≥ , we have F ( s ; ( t + 1) e x ( { } l )) = F ( t ; ( s + 1) e x ( { } l )) . Remark . The case l = 0 in Theorem 1.4 gives Theorem 1.3.2. Proof of Theorem 1.4 In the following, everything defined for indices, such as ζ , will be extended Q -linearly.We denote by ( k ) ˆ ∗ l the formal sum of indices found by adding k to each element of l .For example, if l = ( l , . . . , l r ′ ), we have( k ) ˆ ∗ l = ( l + k, l , . . . , l r ′ ) + · · · + ( l , . . . , l r ′ − , l r ′ + k ) . Note that, throughout this paper, we often use the special case of the harmonic productformula ζ ( k ) ζ ( l ) = ζ (( k ) ∗ l ), where( k ) ∗ l = ( k ) e x l + ( k ) ˆ ∗ l . For an admissible index k , integers s, t ≥ 2, and nonnegative integers m, l , we define F m,l ( s ; k ) := O m (( s ) e x k e x ( { } l )) − O m (( s ) e x ( k e x ( { } l )) † ) ,D m,l ( s, t ) := F m,l ( s ; ( t + 1)) − F m,l ( t ; ( s + 1)) . Note that Theorem 1.4 is equivalent to D m,l ( s, t ) = 0 for any s, t ≥ m, l ≥ D m,l ( s, 2) = 0in the last part of the proof, we use the Hoffman relation (see Theorem 2.6). The proofgoes through the following steps:Lemmas 2.1 and 2.2 ⇒ Lemma 2.3Lemma 2.4 o ⇒ Lemma 2.5Theorem 2.6 ) ⇒ Theorem 1.4 . NOTE ON OHNO SUMS FOR MZVS 3 Lemma 2.1. For integers s ≥ , t ≥ , and m, l ≥ , we have F m,l ( s ; ( t + 1)) = − X m + m = m X | e | = m ζ (( s + m ) ˆ ∗ ((( t + 1) e x ( { } l )) ⊕ e ))+ X m + m = m X | e | = m ζ (( s + m ) ˆ ∗ ((( t + 1) e x ( { } l )) † ⊕ e )) . Proof. By definitions, we have X m + m = m O m ( s ) O m (( t + 1) e x ( { } l ))= X m + m = m X | e | = m ζ (( s + m ) ∗ ((( t + 1) e x ( { } l )) ⊕ e ))= X m + m = m X | e | = m ζ (( s + m ) e x ((( t + 1) e x ( { } l )) ⊕ e ))+ X m + m = m X | e | = m ζ (( s + m ) ˆ ∗ ((( t + 1) e x ( { } l )) ⊕ e ))= O m (( s ) e x ( t + 1) e x ( { } l ))+ X m + m = m X | e | = m ζ (( s + m ) ˆ ∗ ((( t + 1) e x ( { } l )) ⊕ e )) . (1)We also have X m + m = m O m ( s ) O m (( t + 1) e x ( { } l ))= X m + m = m O m ( s ) O m ((( t + 1) e x ( { } l )) † ) (by Theorem 1.2)= X m + m = m X | e | = m ζ (( s + m ) ∗ ((( t + 1) e x ( { } l )) † ⊕ e ))= O m (( s ) e x (( t + 1) e x ( { } l )) † )+ X m + m = m X | e | = m ζ (( s + m ) ˆ ∗ ((( t + 1) e x ( { } l )) † ⊕ e )) . (2)From (1) and (2), we have O m (( s ) e x ( t + 1) e x ( { } l )) − O m (( s ) e x (( t + 1) e x ( { } l )) † )= − X m + m = m X | e | = m ζ (( s + m ) ˆ ∗ ((( t + 1) e x ( { } l )) ⊕ e ))+ X m + m = m X | e | = m ζ (( s + m ) ˆ ∗ ((( t + 1) e x ( { } l )) † ⊕ e )) . By the definition of F m,l ( s ; ( t + 1)), we obtain the result. (cid:3) HIDEKI MURAHARA Lemma 2.2. For integers s, t ≥ , m ≥ , and l ≥ , we have X m + m = m X | e | = m ζ (( s + m ) ˆ ∗ ((( t + 1) e x ( { } l )) ⊕ e )) − X m + m = m − X | e | = m ζ (( s + m + 1) ˆ ∗ ((( t + 1) e x ( { } l )) ⊕ e ))= O m (( s ) ˆ ∗ (( t + 1) e x ( { } l ))) and X m + m = m X | e | = m ζ (( s + m ) ˆ ∗ ((( t + 1) e x ( { } l )) † ⊕ e )) − X m + m = m − X | e | = m ζ (( s + m + 1) ˆ ∗ ((( t + 1) e x ( { } l )) † ⊕ e ))= O m (( s ) ˆ ∗ (( t + 1) e x ( { } l )) † ) . Proof. Since X m + m = m X | e | = m ζ (( l , . . . , l i − , l i + m , l i +1 , . . . , l r ′ ) ⊕ e ) − X m + m = m − X | e | = m ζ (( l , . . . , l i − , l i + m + 1 , l i +1 , . . . , l r ′ ) ⊕ e )= O m ( l , . . . , l r ′ )holds for an admissible index ( l , . . . , l r ′ ) ∈ I and positive integers m, i with 1 ≤ i ≤ r ′ ,we have X m + m = m X | e | = m ζ (( k + m ) ˆ ∗ (( l , . . . , l r ′ ) ⊕ e )) − X m + m = m − X | e | = m ζ (( k + m + 1) ˆ ∗ (( l , . . . , l r ′ ) ⊕ e ))= O m (( k ) ˆ ∗ ( l , . . . , l r ′ ))for a nonnegative integer k . By using this equality, we obtain the first equality. Thesecond equality can be proved in the same way. (cid:3) Lemma 2.3. For integers s ≥ , t ≥ , m ≥ , and l ≥ , we have F m,l ( s − 1; ( t + 1)) − F m − ,l ( s ; ( t + 1))= −O m (( s − 1) ˆ ∗ (( t + 1) e x ( { } l ))) + O m (( s − 1) ˆ ∗ (( t + 1) e x ( { } l )) † ) . NOTE ON OHNO SUMS FOR MZVS 5 Proof. By Lemmas 2.1 and 2.2, we have F m,l ( s − 1; ( t + 1)) − F m − ,l ( s ; ( t + 1))= − X m + m = m X | e | = m ζ (( s + m − 1) ˆ ∗ ((( t + 1) e x ( { } l )) ⊕ e )) − X m + m = m − X | e | = m ζ (( s + m ) ˆ ∗ ((( t + 1) e x ( { } l )) ⊕ e )) ! + X m + m = m X | e | = m ζ (( s + m − 1) ˆ ∗ ((( t + 1) e x ( { } l )) † ⊕ e )) − X m + m = m − X | e | = m ζ (( s + m ) ˆ ∗ ((( t + 1) e x ( { } l )) † ⊕ e )) ! = −O m (( s − 1) ˆ ∗ (( t + 1) e x ( { } l ))) + O m (( s − 1) ˆ ∗ (( t + 1) e x ( { } l )) † ) . This finishes the proof. (cid:3) Lemma 2.4. For integers s, t ≥ and m ≥ , we have O m (( s ) e x (( t ) e x ( { } l )) † ) − O m (( s − 1) ˆ ∗ (( t + 1) e x ( { } l )) † ) − (cid:0) O m (( t ) e x (( s ) e x ( { } l )) † ) − O m (( t − 1) ˆ ∗ (( s + 1) e x ( { } l )) † ) (cid:1) = 0 . Proof. We shall show this lemma only for l ≥ 2. The case l = 1 can be shown similarly.We have( s ) e x (( t ) e x ( { } l )) † = l X i =0 ( s ) e x ( { } i , t, { } l − i ) † = l X i =0 ( s ) e x ( { } i , { } t − , { } l − i +1 )= l X i =0 i X j =0 ( { } j , s, { } i − j , { } t − , { } l − i +1 ) + t − X j =1 ( { } i , { } j , s, { } t − j − , { } l − i +1 )+ l − i X j =0 ( { } i , { } t − , { } j +1 , s, { } l − i − j ) ! and( s − 1) ˆ ∗ (( t + 1) e x ( { } l )) † = l X i =0 ( s − 1) ˆ ∗ ( { } i , t + 1 , { } l − i ) † = l X i =0 ( s − 1) ˆ ∗ ( { } i , { } t − , { } l − i +1 )= l X i =1 i − X j =0 ( { } j , s + 1 , { } i − j − , { } t − , { } l − i +1 ) HIDEKI MURAHARA + l X i =0 t − X j =0 ( { } i , { } j , s, { } t − j − , { } l − i +1 ) + l − i X j =0 ( { } i , { } t − , { } j , s + 1 , { } l − i − j ) ! . Thus, we have O m (( s ) e x (( t ) e x ( { } l )) † ) − O m (( s − 1) ˆ ∗ (( t + 1) e x ( { } l )) † )= l X i =0 i X j =0 O m ( { } j , s, { } i − j , { } t − , { } l − i +1 ) + l − i X j =0 O m ( { } i , { } t − , { } j +1 , s, { } l − i − j ) − l − i X j =0 O m ( { } i , { } t − , { } j , s + 1 , { } l − i − j ) − O m ( { } i , s, { } t − , { } l − i +1 ) ! − l X i =1 i − X j =0 O m ( { } j , s + 1 , { } i − j − , { } t − , { } l − i +1 )= X a + b + c = la,b,c ≥ (cid:0) O m ( { } a , s, { } b , { } t − , { } c +1 ) + O m ( { } a , { } t − , { } b +1 , s, { } c ) − O m ( { } a , { } t − , { } b , s + 1 , { } c ) (cid:1) − X a + b = la,b ≥ O m ( { } a , s, { } t − , { } b +1 ) − X a + b + c = l − a,b,c ≥ O m ( { } a , s + 1 , { } b , { } t − , { } c +1 ) . Similarly, we also have O m (( t ) e x (( s ) e x ( { } l )) † ) − O m (( t − 1) ˆ ∗ (( s + 1) e x ( { } l )) † )= X a + b + c = la,b,c ≥ (cid:0) O m ( { } a , t, { } b , { } s − , { } c +1 ) + O m ( { } a , { } s − , { } b +1 , t, { } c ) − O m ( { } a , { } s − , { } b , t + 1 , { } c ) (cid:1) − X a + b = la,b ≥ O m ( { } a , t, { } s − , { } b +1 ) − X a + b + c = l − a,b,c ≥ O m ( { } a , t + 1 , { } b , { } s − , { } c +1 ) . Then, by using Theorem 1.2, we find the result. (cid:3) Lemma 2.5. For integers s, t ≥ and m ≥ , we have D m − ,l ( s, t ) = D m,l ( s − , t ) + D m,l ( s, t − . Proof. By definitions, we have D m − ,l ( s, t ) − D m,l ( s − , t ) − D m,l ( s, t − F m − ,l ( s ; ( t + 1)) − F m − ,l ( t ; ( s + 1)) − ( F m,l ( s − 1; ( t + 1)) − F m,l ( t ; ( s ))) − ( F m,l ( s ; ( t )) − F m,l ( t − 1; ( s + 1)))= − ( F m,l ( s − 1; ( t + 1)) − F m − ,l ( s ; ( t + 1))) + ( F m,l ( t − 1; ( s + 1)) − F m − ,l ( t ; ( s + 1)))+ F m,l ( t ; ( s )) − F m,l ( s ; ( t )) . NOTE ON OHNO SUMS FOR MZVS 7 Then, by Lemma 2.3, we have D m − ,l ( s, t ) − D m,l ( s − , t ) − D m,l ( s, t − O m (( s − 1) ˆ ∗ (( t + 1) e x ( { } l ))) − O m (( s − 1) ˆ ∗ (( t + 1) e x ( { } l )) † ) − O m (( t − 1) ˆ ∗ (( s + 1) e x ( { } l ))) + O m (( t − 1) ˆ ∗ (( s + 1) e x ( { } l )) † )+ O m (( s ) e x (( t ) e x ( { } l )) † ) − O m (( t ) e x (( s ) e x ( { } l )) † ) . Since O m (( s − 1) ˆ ∗ (( t + 1) e x ( { } l ))) − O m (( t − 1) ˆ ∗ (( s + 1) e x ( { } l )))= O m (( s + t ) e x ( { } l ) + ( s + 1) e x ( t + 1) e x ( { } l − )) − O m (( s + t ) e x ( { } l ) + ( s + 1) e x ( t + 1) e x ( { } l − )) = 0and by Lemma 2.4, we obtain the result. (cid:3) In the proof of Theorem 1.4, we use the following theorem. Theorem 2.6 (Hoffman [2, Theorem 5.1]) . For positive integers k , . . . , k r with k r ≥ ,we have r X i =1 ζ ( k , . . . , k i − , k i + 1 , k i +1 , . . . , k r )= X ≤ i ≤ rk i ≥ k i − X j =0 ζ ( k , . . . , k i − , j + 1 , k i − j, k i +1 , . . . , k r ) . Now we prove our main theorem. Proof of Theorem 1.4. By Lemma 2.5 and the identity D m,l ( s, t ) = − D m,l ( t, s ), we needto show only the case D m,l ( s, 2) = 0, i.e., F m,l ( s ; (3)) = F m,l (2; ( s + 1)). By Lemma 2.1,we have F m,l ( s ; (3)) − F m,l (2; ( s + 1))= X m + m = m X | e | = m (cid:0) − ζ (( s + m ) ˆ ∗ (((3) e x ( { } l )) ⊕ e ))+ ζ (( s + m ) ˆ ∗ (((3) e x ( { } l )) † ⊕ e )) (cid:1) − ( l + 1) O m (( s + 1) e x { } l +1 ) + O m ((2) e x (( s + 1) e x { } l ) † ) . Put A := − X m + m = m X | e | = m ζ (( s + m ) ˆ ∗ (((3) e x ( { } l )) ⊕ e )) ,B := X m + m = m X | e | = m ζ (( s + m ) ˆ ∗ (((3) e x ( { } l )) † ⊕ e )) ,C := − ( l + 1) O m (( s + 1) e x { } l +1 ) + O m ((2) e x (( s + 1) e x { } l ) † ) . Then, by definitions, we have A = − m X a =0 (cid:0) O m − a (( s + a + 3) e x ( { } l )) − O m − a (( s + a + 2) e x (3) e x ( { } l − )) (cid:1) , HIDEKI MURAHARA where we understand the second term is 0 if l = 0, and the same shall apply hereinafter.Note that depths of each MZV in A are all l + 1 and that m X a =0 O m − a ( k , s + a, l ) = X | e | + | e | + u = mu ≥ ( u + 1) × ζ ( k ⊕ e , s + u, l ⊕ e )(3)holds for any nonnegative integers s, m with s ≥ 2, an index k , and an admissible index l . By counting the number of each term, we get A = − X m + ··· + m l +1 = m + s l +1 X p =1 max { m p − s + 1 , } ! × l +1 X i =1 ζ ( m + 2 , . . . , m i − + 2 , m i + 3 , m i +1 + 2 , . . . , m l +1 + 2)(4)(see also Remark 2.7).On the other hand, we have B = X m + m = m l X i =1 i − X j =0 O m ( { } j , s + m + 2 , { } i − j − , , { } l − i +1 )+ l X i =0 O m ( { } i , s + m + 1 , { } l − i +1 )+ l X i =0 l − i X j =0 O m ( { } i , , { } j , s + m + 2 , { } l − i − j ) ! = m X a =0 (cid:0) O m − a ((1) e x ( s + a + 2) e x ( { } l − ) , O m − a (( s + a + 1) e x ( { } l ) , 2) + O m − a ((1) e x ( { } l ) , s + a + 2) (cid:1) . Since O m (( s + 1) e x ( { } l +1 )) = l +1 X i =0 O m ( { } i , s + 1 , { } l − i +1 )and O m ((2) e x (( s + 1) e x ( { } l )) † )= ( l + 2) l +1 X i =0 O m ( { } i , s + 1 , { } l − i +1 ) − O m ( s + 1 , { } l +1 )+ l X i =0 s − X j =1 O m ( { } i , j + 2 , s − j + 1 , { } l − i ) , NOTE ON OHNO SUMS FOR MZVS 9 we also have C = l +1 X i =0 O m ( { } i , s + 1 , { } l − i +1 ) − O m ( s + 1 , { } l +1 )+ l X i =0 s − X j =1 O m ( { } i , j + 2 , s − j + 1 , { } l − i )= O m (2 , ( s + 1) e x ( { } l )) + l X i =0 s − X j =1 O m ( { } i , j + 2 , s − j + 1 , { } l − i )= l X i =0 s − X j =0 O m ( { } i , j + 2 , s − j + 1 , { } l − i ) . Then we find B + C = m X a =0 (cid:0) O m − a ((1) e x ( s + a + 2) e x ( { } l − ) , O m − a (( s + a + 1) e x ( { } l ) , 2) + O m − a ((1) e x ( { } l ) , s + a + 2) (cid:1) + l X i =0 s − X j =0 O m ( { } i , j + 2 , s − j + 1 , { } l − i ) . (5)Note that depths of each MZV in B + C are all l + 2. By using (3) and counting thenumber of each term, we get B + C = X m + ··· + m l +1 = m + s l +1 X p =1 max { m p − s + 1 , } ! × l +1 X i =1 m i X j =0 ζ ( m + 2 , . . . , m i − + 2 , j + 1 , m i − j + 2 , m i +1 + 2 , . . . , m l +1 + 2)(6)(see also Remark 2.8).Hence, from (4) and (6), we find F m,l ( s ; (3)) − F m,l (2; ( s + 1)) = A + B + C = X m + ··· + m l +1 = m + s l +1 X p =1 max { m p − s + 1 , } ! × l +1 X i =1 − ζ ( m + 2 , . . . , m i − + 2 , m i + 3 , m i +1 + 2 , . . . , m l +1 + 2)+ m i X j =0 ζ ( m + 2 , . . . , m i − + 2 , j + 1 , m i − j + 2 , m i +1 + 2 , . . . , m l +1 + 2) ! . By Theorem 2.6, we obtain the result. (cid:3) Remark . To understand (4), we consider the case l = 2. Putting A := O m − a (( s + a + 3 , , , A := O m − a ((3 , s + a + 2 , , A := O m − a ((3 , , s + a + 2)) ,A := O m − a (( s + a + 2 , , , A := O m − a ((2 , s + a + 3 , , A := O m − a ((2 , , s + a + 2)) ,A := O m − a (( s + a + 2 , , , A := O m − a ((2 , s + a + 2 , , A := O m − a ((2 , , s + a + 3)) , and f p := max { m p − s + 1 , } ( p = 1 , , ,g := ζ ( m + 3 , m + 2 , m + 2) , g := ζ ( m + 2 , m + 3 , m + 2) ,g := ζ ( m + 2 , m + 2 , m + 3) , we have A = m X a =0 (cid:0) O m − a (( s + a + 3) e x (2 , O m − a (( s + a + 2) e x (3) e x (2)) (cid:1) = A + · · · + A and X m + m + m = m + s X p =1 max { m p − s + 1 , } ! × X i =1 ζ ( m + 2 , . . . , m i − + 2 , m i + 3 , m i +1 + 2 , . . . , m l +1 + 2)= X m + m + m = m + s ( f g + · · · + f g ) . By seeing the correspondences A ↔ f g , A ↔ f g , A ↔ f g ,A ↔ f g , A ↔ f g , A ↔ f g ,A ↔ f g , A ↔ f g , A ↔ f g , we find A + · · · + A = P m + m + m = m + s ( f g + · · · + f g ). Remark . To understand (6), here we consider two examples: the cases ( l, m ) = (0 , l, m ) = (1 , l, m ) = (0 , X a =0 (cid:0) O − a (1 , s + a + 2) + O − a ( s + a + 1 , (cid:1) + s − X j =0 O ( j + 2 , s − j + 1) . NOTE ON OHNO SUMS FOR MZVS 11 By using (3), we find that the coefficients of each MZV are all 3. Indeed, we haveR.H.S. of (5)= (cid:0) O (1 , s + 4) + O ( s + 3 , (cid:1) + (cid:0) O (1 , s + 3) + O ( s + 2 , (cid:1) + (cid:0) O (1 , s + 2) + O ( s + 1 , (cid:1) + s − X j =0 O ( j + 1 , s − j + 2)= (cid:0) ζ (1 , s + 4) + ζ ( s + 3 , (cid:1) + (cid:0) ζ (1 , s + 4) + ζ (2 , s + 3) + ζ ( s + 2 , 3) + ζ ( s + 3 , (cid:1) + (cid:0) ζ (1 , s + 4) + ζ (2 , s + 3) + ζ (3 , s + 2) + ζ ( s + 1 , 4) + ζ ( s + 2 , 3) + ζ ( s + 3 , (cid:1) + s − X j =0 (cid:0) ζ ( j + 1 , s − j + 4) + ζ ( j + 2 , s − j + 3) + ζ ( j + 3 , s − j + 2) (cid:1) = 3 s +2 X j =0 ζ ( j + 1 , s − j + 4) = R.H.S. of (6) . Similarly, when ( l, m ) = (1 , X a =0 (cid:0) O − a (1 , s + a + 2 , α + O − a ( s + a + 2 , , O − a (2 , s + a + 1 , β + O − a ( s + a + 1 , , O − a (1 , , s + a + 2) + O − a (2 , , s + a + 2) (cid:1) + s − X j =0 (cid:0) O ( j + 2 , s − j + 1 , γ + O (2 , j + 2 , s − j + 1) δ (cid:1) . By (3), we also find the equality (6). For example, one can find the coefficient of ζ (2 , s +2 , 3) (for arbitrary s ) from the underlined terms, i.e., α ( a = 0) → β ( a = 0 , → γ → 1, and δ → 1. The sum of these numbers coincide with the sum coming from ǫ → η → (cid:0) O (1 , s + 4 , 2) + O ( s + 4 , , 2) + O (2 , s + 3 , 2) + O ( s + 3 , , O (1 , , s + 4) + O (2 , , s + 4) (cid:1) + (cid:0) O (1 , s + 3 , 2) + O ( s + 3 , , 2) + O (2 , s + 2 , 2) + O ( s + 2 , , O (1 , , s + 3) + O (2 , , s + 3) (cid:1) + (cid:0) O (1 , s + 2 , 2) + O ( s + 2 , , 2) + O (2 , s + 1 , 2) + O ( s + 1 , , O (1 , , s + 2) + O (2 , , s + 2) (cid:1) + s − X j =0 (cid:0) O ( j + 2 , s − j + 1 , 2) + O (2 , j + 2 , s − j + 1) (cid:1) = (cid:0) ζ (1 , , s + 4) + 2 ζ (1 , , s + 3) + ζ (1 , , s + 2) + ζ (1 , s + 2 , ζ (1 , s + 3 , 3) + 3 ζ (1 , s + 4 , 2) + 3 ζ (2 , , s + 4) + 4 ζ (2 , , s + 3) + 2 ζ (2 , , s + 2) + ζ (2 , s + 1 , 4) + 3 ζ (2 , s + 2 , 3) + 5 ζ (2 , s + 3 , ζ (3 , , s + 3) + 2 ζ (3 , , s + 2) + ζ (3 , s + 1 , 3) + 3 ζ (3 , s + 2 , ζ (4 , , s + 2) + ζ (4 , s + 1 , 2) + ζ ( s + 1 , , 4) + ζ ( s + 1 , , ζ ( s + 1 , , 2) + ζ ( s + 2 , , 4) + 3 ζ ( s + 2 , , 3) + 3 ζ ( s + 2 , , ζ ( s + 3 , , 3) + 5 ζ ( s + 3 , , 2) + 3 ζ ( s + 4 , , (cid:1) + s − X j =0 (cid:0) O ( j + 2 , s − j + 1 , 2) + O (2 , j + 2 , s − j + 1) (cid:1) = X m + m = s +2 max { m − s + 1 , } + max { m − s + 1 , } ! × m X j =0 ζ ( j + 1 , m − j + 2 , m + 2) ǫ + m X j =0 ζ ( m + 2 , j + 1 , m − j + 2) η ! = R.H.S. of (6) . References [1] M. Hirose, H. Murahara, T. Onozuka, and N. Sato, Linear relations of Ohno sums of multiple zetavalues , Indag. Math. (2020), 556–567.[2] M. E. Hoffman, Multiple harmonic series , Pacific J. Math. (1992), 275–290.[3] Y. Ohno, A generalization of the duality and sum formulas on the multiple zeta values , J. NumberTheory (1999), 39–43.(Hideki Murahara) Nakamura Gakuen University Graduate School, 5-7-1, Befu, Jonan-ku, Fukuoka, 814-0198, Japan Email address ::