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seki@math.aoyama.ac.jp Department of Mathematical We introduce the multivariable connected sum which is a generalization of Sciences, Aoyama Gakuin Seki–Yamamoto’s connected sum and prove the fundamental identity for these sums University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara-shi, by series manipulation. This identity yields explicit procedures for evaluating Kanagawa 252-5258, Japan multivariable connected sums and for giving relations among special values of multiple Full list of author information is polylogarithms. In particular, our class of relations contains Ohno’s relations for multiple available at the end of the article This research was supported in polylogarithms. part by JST Global Science Keywords: Multiple zeta values, Multiple polylogarithms, Connected sums, Connector, Campus ROOT program and by JSPS KAKENHI Grant Numbers Ohno’s relation 18J00151, 21K13762 1 Introduction In [13], a symmetrical double series expression of ζ (2) due to B. Cloitre is exhibited: ∞ ∞ (m − 1)!(n − 1)! ζ (2) = . (1.1) (m + n)! m=1 n=1 The right-hand side of (1.1) is a special case of the connected sum introduced by the third author and Yamamoto [12]. We provide a brief review of this theory here. We call a tuple of positive integers k = (k , ... ,k )an index.If k ≥ 2, then we call k an 1 r r admissible index. We call r (resp. k + ··· + k )the depth (resp. weight)of k and denote 1 r it by dep(k) (resp. wt(k)). We regard the empty tuple ∅ as an admissible index and call it the empty index.Weset dep(∅):=0 and wt(∅):=0. For two indices k = (k , ... ,k )and 1 r l = (l , ... ,l ), the connected sum Z(k; l)isdeﬁnedby 1 s ⎡ ⎤ r s m !n ! 1 1 r s ⎣ ⎦ Z(k; l):= · · ∈ R ∪{+∞}. k l i j (m + n )! r s m 0=m <m <···<m i=1 j=1 0 1 r i 0=n <n <···<n 0 1 s Throughout, we understand that the empty product (resp. sum) is 1 (resp. 0). It is straightforward to see that the symmetry Z(k; l) = Z(l; k) and the boundary condition Z(k; ∅) = ζ (k)hold. Here, ζ (k)isthe multiple zeta value (MZV) ζ (k):= k k 1 r m ··· m 0=m <m <···<m 1 0 1 r and this series is convergent if k is admissible. The most important properties for the connected sum are the transport relations. To state them, we introduce arrow notation for © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV 4 Page 2 of 25 Kawamura et al. Res Math Sci (2022) 9:4 indices. For a non-empty index k = (k , ... ,k ), we deﬁne k (resp. k )tobe(k , ... ,k , 1) 1 r → ↑ 1 r (resp. (k , ... ,k ,k + 1)). We set ∅ :=(1) and ∅ :=∅. Then, the transport relations 1 r−1 r → ↑ for Z(k; l) are as follows (the q → 1, x → 0caseof[12, Theorem 2.2]): Z(k ; l) = Z(k; l )(l = ∅), → ↑ (1.2) Z(k ; l) = Z(k; l )(k = ∅). ↑ → By using these properties of the connected sum, we can prove the duality relation for MZVs † † “ζ (k) = ζ (k ).” Here, we explain the deﬁnition of the dual index k . For a non-negative integer m and a complex number z, the symbol {z} denotes m repetitions of z.Any a −1 a −1 admissible index k = ∅ is uniquely expressed as k = ({1} ,b + 1, ... , {1} ,b + 1 h 1), where h, a , ... ,a ,b , ... ,b are positive integers. Then, the dual index is deﬁned 1 h 1 h † b −1 b −1 † as k :=({1} ,a + 1, ... , {1} ,a + 1). For the empty index, we use ∅ :=∅.The h 1 duality relation is immediately proved by the change of variables in the iterated integral representation of the MZV (a special case of (1.6) below). We emphasize that Seki– Yamamoto’s proof is executed using only series manipulation and no iterated integral is required. Since the single index k = (2) is self-dual, the duality for this k is a tautology “ζ (2) = ζ (2).” However, the transportation process includes a non-trivial component as follows: ζ (2) = Z(2; ∅) = Z(1; 1) = Z(∅;2) = ζ (2). (1.3) We see that the identity ζ (2) = Z(1; 1) is nothing but Cloitre’s identity (1.1). Note that parentheses will often be omitted, such as Z((2); ∅) = Z(2; ∅)and Z((1); (1)) = Z(1; 1). In [14], an elegant series expression of Apéry’s constant ζ (3) owing to O. Oloa is exhib- ited: ∞ ∞ m+n 1 (m − 1)!(n − 1)! 1 ζ (3) = . (1.4) 3 (m + n)! j m=1 n=1 j=1 Identities (1.1)and (1.4) are also discussed on the website “les-mathematiques.net” in a forum post entitled “Accélération pour zeta(2)” [1]. In a post entitled “Série triple” [10] on the same website, the following series identity is given: ∞ ∞ ∞ (m − 1)!(m − 1)!(m − 1)! 13 π 1 2 3 = ζ (3) − log 2. (m + m + m )! 4 2 1 2 3 m =1 m =1 m =1 1 2 3 Furthermore, a user named Amtagpa showed ∞ ∞ (m − 1)! ··· (m − 1)! 1 1 1 n+1 ··· = n!Li 1, , ... , (1.5) n−1 {1} ,2 (m + ··· + m )! 2 n 1 n+1 m =1 m =1 1 n+1 for a general positive integer n by using the iterated integral representation. Here, the multiple polylogarithm (MPL) Li (z , ... ,z ) is of the so-called shuﬄe-type: 1 r r m −m i i−1 X i Li (z , ... ,z ):= , 1 r 0=m <m <···<m i=1 0 1 r i where k = (k , ... ,k )isanindex and z , ... ,z are complex numbers satisfying 1 r 1 r |z |, ... , |z |≤ 1. If (k , |z |) = (1, 1), then this series is absolutely convergent. The shuﬄe- 1 r r r type MPL has the iterated integral representation X r −1 −1 Li (z , ... ,z ) = (−1) I (z , ... ,z ) (1.6) 1 r k k 1 r Kawamura et al. Res Math Sci (2022) 9:4 Page 3 of 25 4 in the notation of [5]. The shuﬄe-type MPLs inherit the shuﬄe product property of iterated integrals directly in the z variables, and this fact is the reason why we call these series “shuﬄe-type.” There is another type of MPL, the harmonic-type MPL used and deﬁned in Sect. 4, but we principally focus on the shuﬄe-type MPL in this paper. We can X 1 13 π check that 2Li 1, = ζ (3) − log 2 holds. For example, by the duality for MPLs 1,2 2 4 2 X 1 (the n = 2caseof(5.1)), we see that Li 1, =−ζ (1, 2); see Sect. 3.2 for the notation for 1,2 2 13 π the alternating multiple zeta values. Then, the evaluation −2ζ (1, 2) = ζ (3) − log 2 4 2 is conﬁrmed by [16, Proposition 14.2.7]. Our focus in this paper is on interpreting identities (1.4)and (1.5) by transporting indices in the same way as in Cloitre’s identity (1.1). For this purpose, we extend Seki– Yamamoto’s connected sum to the multiple connected sum as follows. We set S :={m = r+1 (m ,m , ... ,m ) ∈ Z | m = 0 < m < ··· < m } and S (m):={q = (q ,q , ... ,q ) ∈ 0 1 r 0 1 r 0 1 s s+1 Z | q = 0 < q ≤ ··· ≤ q = m} for non-negative integers r, s and m. Furthermore, 0 1 s we set (a )!(a )! ··· (a )! 1 2 n C(a ,a , ... ,a ):= 1 2 n (a + a + ··· + a )! 1 2 n for a positive integer n and non-negative integers a , ... ,a .Thisvalue C(a ,a , ... ,a ) 1 n 1 2 n is just the inverse of a multinomial coeﬃcient, but in the context of multiple connected sums, we call it the connector because of its role. Using these notation, for a positive integer (1) (1) (n) (n) n and indices k = (k , ... ,k ), ... , k = (k , ... ,k )and l = (l , ... ,l ), we deﬁne 1 r n r 1 s 1 1 1 n the multiple connected sum Z (k ; ... ; k | l)as n 1 n (1) (2) (n) Z (k ; ... ; k | l):= C(m ,m , ... ,m ) n 1 n r r r 1 2 n (1) (n) m ∈S ,...,m ∈S r r (1) (2) (n) q∈S (m +m +···+m ) r r r s n 1 2 ⎡ ⎛ ⎞ ⎤ n s 1 1 ⎣ ⎝ ⎠ ⎦ · · q , (j) (j) i (m ) j=1 i=1 i=1 (1) (1) (n) (n) (1) (n) where m = (m , ... ,m ), ... , m = (m , ... ,m )and q = (q , ... ,q ). Set r r 1 s 1 1 1 n Z (k ; ... ; k | ∅):=0and Z (k ; ... ; k ):=Z (k ; ... ; k | 1). n 1 n n 1 n n 1 n The original Seki–Yamamoto’s connected sum Z(k; l) coincides with our Z (k; l) = Z (k; l | 1), not Z (k | l). The series appearing in identities (1.4)and (1.5)are Z (1; 1 | 1, 1) 2 1 2 and Z (1; ... ; 1), respectively. The presence of the l-component in the deﬁnition is n+1 advantageous for obtaining relations among MZVs or MPLs, and the n = 1caseofour connected sum has been explored in previous studies; Z (k | l )(k, l = ∅) is the multiple 1 ↑ zeta value of Kaneko–Yamamoto type introduced in [6], and Z (k | l )(k is admissible) 1 → is the Schur multiple zeta value of anti-hook type introduced in [7]. Note that these two kinds of series are the same object by deﬁnition or by the n = 1caseofProposition 2.2 below. In what follows, we provide an explicit procedure for calculating Z (k ; ... ; k | l) n 1 n via transportations of indices and prove the transport relations by series manipulation. In addition to identities (1.4)and (1.5), our method leads to a formula for ζ (4) (see Example 5.4): ∞ ∞ m+n 8 (m − 1)!(n − 1)! 1 ζ (4) = . (1.7) 17 (m + n)! j m=1 n=1 j=1 4 Page 4 of 25 Kawamura et al. Res Math Sci (2022) 9:4 More generally, we prove the following. Theorem 1.1 Let n be an integer at least 2 and k , ... , k , l non-empty indices. We set 1 n k:=wt(k ) + ··· + wt(k ) + wt(l) and deﬁne MPL(n, k) by 1 n 1≤r≤k−2,k is admissible with wt(k)=k−1, MPL(n, k):= Li (1,z , ... ,z ) 1 1 . 2 r k dep(k)=r,and z ,...,z ∈{1, ,..., } 2 r 2 n Then, the multiple connected sum Z (k ; ... ; k | l) can be written explicitly as a Z-linear n 1 n combination of elements of MPL(n, k). The proof of this theorem by transporting indices requires the use of more general connected sums; see Example 3.3 for instance. In view of this phenomenon, it is nat- ural to deﬁne the following generalization of the multiple connected sum. Let n be a (1) (1) (n) (n) positive integer and k = (k , ... ,k ), ... , k = (k , ... ,k )and l = (l , ... ,l ) 1 r n r 1 s 1 1 1 n (1) (1) (n) (n) indices. Let z = (z , ... ,z ), ... , z = (z , ... ,z )and w = (w , ... ,w )beele- 1 r n r 1 s 1 1 1 n r r s 1 n ments of D , ... , D and D , respectively. Here, the symbol D denotes the unit disk {z ∈ C ||z|≤ 1} and the unique element of D is denoted by ∅. Then, the multivariable connected sum is deﬁned by z z 1 n w (1) (2) (n) Z ; ... ; := C(m ,m , ... ,m ) k k l r r r 1 n 1 2 n (1) (n) m ∈S ,...,m ∈S r r (1) (2) (n) q∈S (m +m +···+m ) r r r s n 1 2 ⎡ ⎛ ⎞ ⎤ (j) (j) j (j) q −q n s m −m i i−1 i i−1 (z ) w i i ⎣ ⎝ ⎠ ⎦ · · q , (j) (j) i (m ) j=1 i=1 i=1 (1) (1) (n) (n) (1) (n) where m = (m , ... ,m ), ... , m = (m , ... ,m )and q = (q , ... ,q ). Set r r 1 s 1 1 1 n z z z z z z 1 n ∅ 1 n 1 n Z ; ... ; :=0and Z ; ... ; :=Z ; ... ; . We establish the fundamen- n n n k k ∅ k k k k 1 1 n 1 n 1 n tal identity for multivariable connected sums. To do so, we generalize the arrow notation dep(k) appropriately. Let k be an index and z ∈ D .Let v be an element of D \{0}. Then, we use the following notation: z (z,v) z z z z v := , := := , ∞ :=− . 0 k k k k k ↑ k k −→ ↑ ↑ → −→ −→ In relation to the last symbol, we use the following deﬁnition: z z z z 1 n w 1 n w Z ; ... ; := ··· · Z ; ... ; n 1 n l 1 n n l k k k k 1 n 1 n for , ... , ∈{1, −1}. 1 n The following identity is key to this paper. Theorem 1.2 (Fundamental identity) Let n be an integer greater than 1 and k , ... , k 1 n+1 dep(k ) indices. Take tuples of complex numbers z ∈ D for all 1 ≤ i ≤ n + 1. Furthermore, take v , ... ,v ∈ (D \{0}) ∪{∞} and v ∈ D satisfying 1 n n+1 1 1 + ··· + = v n+1 v v 1 n and the assumption that (1) if k = ∅ (1 ≤ i ≤ n),thenv =∞,and i i (2) if n = 2 and k = ∅ for 1 ≤ i ≤ 2,then |v | = 1 or |v | < 1. i 3−i 3 Kawamura et al. Res Math Sci (2022) 9:4 Page 5 of 25 4 Here, we understand 1/∞= 0 and |∞|=∞. Then, we have n+1 z z z 1 i n n+1 Z v ; ... ; ; ... ; vn v = 0. n 1 k k k k n+1 1 i n n+1 −→ −→ −→ i=1 z z z z n+1 1 i n Here, the symbol Z v ; ... ; ; ... ; v means that only the ith com- n v n 1 k k k k n+1 1 i n n+1 −→ −→ −→ ponent is not equipped with the arrow notation. For example, the n = 3 case for the fundamental identity is z z z z 1 2 3 4 v v v Z ; ; 3 2 3 4 k k k k 1 2 3 4 −→ −→ −→ z z z z 1 2 3 4 + Z v ; ; v v 3 1 3 4 k k k k 1 2 3 4 −→ −→ −→ z z z z 1 2 3 4 + Z v ; v ; v 3 1 2 4 k k k k 1 2 3 4 −→ −→ −→ z z z z 1 2 3 4 + Z v ; v ; v 3 1 2 3 k k k k 1 2 3 4 −→ −→ −→ = 0. The fundamental identity supplies all needed transport relations for the multivariable connected sums in this paper. By transporting indices and variables algorithmically, in accordance with the method presented in [11], a given multivariable connected sum satisfying a suitable condition is written as a Z-linear combination of Z -values. We regard the boundary conditions for multivariable connected sums as the fact that Z -values are written as Z-linear combinations of MPLs; if w = l = (1), then the boundary conditions are trivial. Thus, main theorems of the present paper are Theorem 1.1,Theorem 1.2 and the following two theorems: Theorem 1.3 (see Theorem 5.2 for the precise statement) Every multivariable connected sum with transportable variables can be expressed explicitly as a Z-linear combination of absolutely convergent multiple polylogarithms. Theorem 1.4 (see Sect. 6.1 for the precise recipe) Transport relations and boundary conditions for multivariable connected sums give a family of functional relations among multiple polylogarithms. As a special case, Ohno’s relation for multiple polylogarithms is obtained. For simplicity, conditional convergent series and analytic continuations are disregarded herein. The remainder of the paper is organized as follows. In Sect. 2, we prove the fundamental identity. In Sect. 3, we explain transport relations derived from the fundamental identity and reprove the duality relation for MPLs. In Sect. 4, we describe the boundary conditions. In Sect. 5, we give the recipe for obtaining evaluations of multivariable connected sums and prove Theorem 1.1 and Theorem 1.3. Finally, in Sect. 6, we give the recipe for obtaining relations among MPLs and prove Theorem 1.4. Ohno’s relation for MPLs is also proved here. Some examples are exhibited in Sects. 3–6. 2 The fundamental identity First, we prepare basic properties of the multivariable connected sum. 4 Page 6 of 25 Kawamura et al. Res Math Sci (2022) 9:4 Proposition 2.1 Let n be a positive integer and k , ... , k , l indices. Take tuples of complex 1 n dep(k ) dep(l) numbers z ∈ D for all 1 ≤ i ≤ nand w ∈ D . Then, the following statements hold: (i) For z = (z , ... ,z ), w = (w , ... ,w )(r = dep(k ),s = dep(l)), we assume that 1 1 r 1 s 1 1 w |z | < 1 or |w | < 1 holds if both k and l are non-admissible. Then, Z r s 1 1 k l 1 X converges absolutely, and for the case of l = (1), w = (1),wehaveZ = Li (z ). 1 1 1 k (ii) If n > 1 and all k are non-empty, then the multivariable connected sum z z 1 n w Z ; ... ; k k l 1 n converges absolutely. (iii) For any permutation σ of degree n, z z z z σ (1) σ (n) 1 n w w Z ; ... ; = Z ; ... ; n l n l k k k k 1 n σ (1) σ (n) holds. (iv) Let n > 1.For all 1 ≤ i ≤ n, we have z z z q z i−1 i+1 z z z 1 ∅ n w 1 i n w Z ; ... ; ; ; ; ... ; = Z ; ... ; ; ... ; , n n−1 k k ∅ k k l k q k l 1 i−1 i+1 n 1 n where denotes the skipped entry. Proof We prove only (ii) because the others are obvious by deﬁnition. It is suﬃcient to show that Z (k ; ... ; k | l) converges. Since n > 1, the inequality n 1 n C(m , ... ,m ) ≤ (2.1) 1 n m ··· m 1 n holds for positive integers m , ... ,m ;the case n = 2 is proved from the estimate 1 n m + m m + m m m 1 2 1 2 1 2 ≥ = + + m m > m m for m ≥ m ≥ 2, 1 2 1 2 1 2 m 2 2 2 and then the general case follows from induction on n. Take any ε> 0and set s = dep(l) and l = (l , ... ,l ). Then, there exists some positive constant c depending on ε,s,n 1 s ε,s,n such that ≤ c · (m ··· m ) . ε,s,n 1 n l l 1 s q ··· q 1≤q ≤···≤q =m +···+m 1 s 1 n 1 By these elementary facts, we see that Z (k ; ... ; k | l) is bounded above by a constant n 1 n multiple of a product of n convergent special values of the multiple zeta function. By virtue of Proposition 2.1 (iii), Theorem 1.2 is a consequence of Proposition 2.2 and Theorem 2.4 below. Proposition 2.2 Let n be a positive integer. Let k , ... , k , l be non-empty indices. Take 1 n dep(k ) dep(l) tuples of complex numbers z ∈ D for all 1 ≤ i ≤ nand w ∈ D . Then, we have z z z z z w w 1 i n 1 n Z ; ... ; ; ... ; = Z ; ... ; . ( ) n n k k k l ↑ k k l 1 ↑ i n ↑ 1 ↑ n ↑ i=1 Proof This follows from the identity m m + ··· + m i 1 n m ··· m m ··· m 1 n 1 n i=1 for positive integers m , ... ,m . 1 n Kawamura et al. Res Math Sci (2022) 9:4 Page 7 of 25 4 For a convergence argument, we need the following lemma. Lemma 2.3 Let n and N be integers satisfying N ≥ n ≥ 2. Then, we have C(a , ... ,a ) 1 1 n = O . a ··· a N 1 n (a ,...,a )∈Z 1 n >0 a +···+a =N 1 n 1 2 2 Proof By estimates ≤ = and ab a+b N N −1 N −2 1 2 1 2 2(N − 3) 4 C(a, b) = ≤ + = + ≤ , N N N N N(N − 1) N a=1 a a=2 2 (a,b)∈Z >0 a+b=N the statement for n = 2 is true. Next, we assume that n ≥ 3 and the desired estimate for n − 1 holds. Then, there exists some positive constant c such that we have N −n+1 C(a , ... ,a ) C(N − a ,a ) C(a , ... ,a ) 1 n n n 1 n−1 a ··· a a a ··· a 1 n n 1 n−1 n−1 a =1 (a ,...,a )∈Z n 1 n (a ,...,a )∈Z >0 1 n−1 >0 a +···+a =N 1 n a +···+a =N −a 1 n−1 n N −n+1 C(N − a ,a ) c n n ≤ · a (N − a ) n n a =1 N −1 c C(N − a ,a ) c 8 n n ≤ ≤ · . n − 1 a (N − a ) n − 1 N n n a =1 Since ≤ 1 for n ≥ 9, we can take a constant in the big-O notation to be independent n−1 of n. This proves the lemma. Theorem 2.4 Let n and d be integers satisfying n ≥ 2 and 1 ≤ d ≤ n. Let k , ... , k , l be dep(k ) indices and k , ... , k non-empty indices. Take tuples of complex numbers z ∈ D n i d+1 dep(l) for all 1 ≤ i ≤ n, w ∈ D and complex numbers v , ... ,v ∈ D \{0},t ∈ D satisfying 1 1 + ··· + = t (2.2) v v 1 d and the assumption that if n = d = 2 and k = ∅ (1 ≤ i ≤ 2),then |v | < 1 or |t| < 1. i 3−i Then, we have z z z z d+1 z 1 i d n w v v Z ; ... ; ; ... ; ; ; ... ; ( ) n 1 t k k k d k k l 1 i n d d+1 ↑ −→ −→ −→ i=1 z z z d+1 z z 1 d i n w v v − Z ; ... ; ; ; ... ; ; ... ; ( ) n 1 t k k d k k k l 1 i n d d+1 ↑ −→ −→ −→ i=d+1 z z z 1 d d+1 n w =−Z v ; ... ; v ; ; ... ; ( ) . n 1 d l k k k k 1 d d+1 n ↑ −→ −→ ↑ 4 Page 8 of 25 Kawamura et al. Res Math Sci (2022) 9:4 Proof For vectors x = (x , ... ,x )and y = (y , ... ,y ), we use the following notation in 1 d 1 d this proof: def x ≺ y ⇐⇒ x < y for all 1 ≤ j ≤ d, j j def x ≺ y ⇐⇒ x < y for all 1 ≤ j ≤ d, j = i and x = y , i j j i i def x y ⇐⇒ x < y for all 1 ≤ j ≤ d, j = i and x ≤ y . i j j i i Then, it is suﬃcient to show that an identity ⎡ ⎤ d d a −m k k ⎢ ⎥ 1 + + k |a|+|m |−q ⎢ ⎥ C(a , ... ,m , ... ,a , m ) · · t 1 i d ⎣ ⎦ m ··· m a d+1 n k d − i=1 k=1 a∈Z ,m ≺ a + k=i q≤|a|+|m | ⎡ ⎤ a −m n d k k m + + k |a|+|m |−q ⎣ ⎦ − C(a , ... ,a , m ) · · t m ··· m a d+1 k d − i=d+1 k=1 a∈Z ,m ≺a q≤|a|+|m | ⎡ ⎤ a −m k k + k ⎣ ⎦ =− C(a , ... ,a , m ) · · q m ··· m a d+1 k d − k=1 a∈Z ,m ≺a q=|a|+|m | holds for non-negative integers m , ... ,m ,q and positive integers m , ... ,m . Here, 1 d d+1 n − + we abbreviated (m , ... ,m ) (resp. (m , ... ,m )) to m (resp. m ). We also abbre- 1 d d+1 n viated (a , ... ,a )to a for variables. The symbol |a|+|m | denotes a + ··· + a + 1 d 1 d m + ··· + m . Note that C(a , ... ,a , m )means C(a , ... ,a ,m , ... ,m ), not d+1 n 1 d 1 d d+1 n C(a , ... ,a , (m , ... ,m )). We see that each series above converges absolutely from 1 d d+1 n the estimate (2.1) or the assumption for the case where n = d = 2. Note that we cover the 0 + case where t = 0 by considering 0 as 1. For the d = n case, we understand that m = ∅ and m ··· m = 1. For the d = 1 case, we understand the ﬁrst term of the left-hand d+1 n side of the above identity as m +···+m −q 1 n · C(m , ... ,m ) · t 1 n m ··· m 2 n when q ≤ m + ··· + m and 0 otherwise. 1 n Take a suﬃciently large integer N. For each 1 ≤ i ≤ d, ⎡ ⎤ d a −m k k ⎢ v ⎥ + k |a|+|m |−q ⎢ ⎥ C(a , ... ,m , ... ,a , m ) · · t 1 i ⎣ ⎦ d − k=1 a∈Z ,m ≺ a k=i q≤|a|+|m |<N ⎡ ⎤ a −m k k v + + k |a|+|m |−q ⎣ ⎦ = C(a , ... ,a , m ) · a · t 1 d i d − k=1 a∈Z ,m a q≤|a|+|m |<N ⎡ ⎤ d a −m k k + k |a|+|m |−q ⎣ ⎦ − C(a , ... ,a , m ) · a · t . 1 d i d − k=1 a∈Z ,m ≺a q≤|a|+|m |<N Kawamura et al. Res Math Sci (2022) 9:4 Page 9 of 25 4 For the ﬁrst term of the right-hand side, by replacing a with a − 1, i i ⎡ ⎤ a −m k k v + + k |a|+|m |−q ⎣ ⎦ C(a , ... ,a , m ) · a · t 1 d i d − k=1 a∈Z ,m a q≤|a|+|m |<N ⎡ ⎤ d a −m k k a + + k |a|+|m |−q−1 ⎣ ⎦ = C(a , ... ,a − 1, ... ,a , m ) · · t 1 i d v a i k d − k=1 a∈Z ,m ≺a q<|a|+|m |≤N ⎡ ⎤ d a −m k k 1 + + + k |a|+|m |−q−1 ⎣ ⎦ = (|a|+|m |) · C(a , ... ,a , m ) · · t . 1 d v a i k d − k=1 a∈Z ,m ≺a q<|a|+|m |≤N By summing from i = 1to d and using (2.2), we have ⎡ ⎤ d d a −m k k + |a|+|m |−q ⎣ ⎦ C(a , ... ,a , m ) · a · t 1 d i d − i=1 k=1 a∈Z ,m a q≤|a|+|m |<N ⎡ ⎤ a −m k k + + k |a|+|m |−q ⎣ ⎦ = (|a|+|m |) · C(a , ... ,a , m ) · · t . d − k=1 a∈Z ,m ≺a q<|a|+|m |≤N On the remaining terms, ⎡ ⎤ d d a −m k k 1 + + k |a|+|m |−q ⎣ ⎦ C(a , ... ,a , m ) · a · t 1 i m ··· m a d+1 k d − i=1 k=1 a∈Z ,m ≺a q≤|a|+|m |<N ⎡ ⎤ n d a −m k k m + + k |a|+|m |−q ⎣ ⎦ + C(a , ... ,a , m ) · · t m ··· m a d+1 n k d − i=d+1 k=1 a∈Z ,m ≺a q≤|a|+|m |<N + + = (|a|+|m |) · C(a , ... ,a , m ) m ··· m d+1 d − a∈Z ,m ≺a q≤|a|+|m |<N ⎡ ⎤ a −m k k k |a|+|m |−q ⎣ ⎦ · · t . k=1 By combining these calculations, we have ⎡ ⎤ a −m d d k k ⎢ ⎥ 1 v + + k |a|+|m |−q ⎢ ⎥ C(a , ... ,m , ... ,a , m ) · · t 1 i d ⎣ ⎦ m ··· m a d+1 n k i=1 d − k=1 a∈Z ,m ≺ a k=i q≤|a|+|m |<N ⎡ ⎤ n d a −m k k m + + k |a|+|m |−q ⎣ ⎦ − C(a , ... ,a , m ) · · t 1 d m ··· m a d+1 n k d − i=d+1 k=1 a∈Z ,m ≺a q≤|a|+|m |<N 4 Page 10 of 25 Kawamura et al. Res Math Sci (2022) 9:4 + + = (|a|+|m |) · C(a , ... ,a , m ) m ··· m d+1 n d − a∈Z ,m ≺a q<|a|+|m |≤N ⎡ ⎤ d a −m k k k |a|+|m |−q ⎣ ⎦ · · t k=1 + + − (|a|+|m |) · C(a , ... ,a , m ) 1 d m ··· m d+1 n d − a∈Z ,m ≺a q≤|a|+|m |<N ⎡ ⎤ a −m k k v + k |a|+|m |−q ⎣ ⎦ · · t k=1 ⎡ ⎤ d a −m k k + k ⎣ ⎦ =− C(a , ... ,a , m ) · · q 1 d m ··· m a d+1 n k d − k=1 a∈Z ,m ≺a q=|a|+|m | ⎡ ⎤ d a −m k k + k N −q ⎣ ⎦ + C(a , ... ,a , m ) · · t . 1 d m ··· m a d+1 n k d − k=1 a∈Z ,m ≺a |a|+|m |=N The last term tends to 0 as N tends to inﬁnity by Lemma 2.3. This completes the proof. 3 Transport relations Let m be a positive integer. For t ∈ D,wedeﬁne B (t)by m m 1 1 B (t):= (v , ... ,v ) ∈ ((D \{0}) ∪{∞}) = t or t − ≥ 1 . m 1 m v v i i i=1 i=1 For example, B (1) ={z ∈ D \{0}| Re(z) ≤ }∪{1, ∞}, where Re(z) denotes the real part of z. dep(k ) Let n be an integer greater than 1. Let k , ... , k , l be indices and z ∈ D , ... , z ∈ 1 n 1 n dep(k ) dep(l) D , w ∈ D .Let t ∈ D and (v , ... ,v ) ∈ B (t). We assume that 1 n−1 n−1 •if k = ∅ (1 ≤ i ≤ n − 1), then v =∞, i i n−1 1 •if k = ∅, then = t, i=1 •if n = 2, k = ∅ and |t|= 1, then t − = 1and •if n = 2, k = ∅ and |t|= 1, then |v | = 1. 2 1 In this setting, by the fundamental identity, we have z z 1 n−1 n w Z v ; ... ; v ; ( ) n 1 t k k n−1 k l 1 n−1 n −→ −→ −→ n−1 z z z 1 i n w =− Z v ; ... ; ; ... ; v ( ) n 1 n t k k k l 1 n i −→ −→ −→ i=1 z z w 1 n − Z v ; ... ; v , (3.1) n 1 k k l 1 n −→ −→ 1 n−1 1 where v ∈ (D\{0})∪{∞} is deﬁned by = t − . In this equality, the total weight i=1 v v n i of indices (including the arrow parts) except for the nth index of each multivariable con- nected sum appearing in the right-hand side is 1 less than the total weight of indices except Kawamura et al. Res Math Sci (2022) 9:4 Page 11 of 25 4 for the nth index of the multivariable connected sum in the left-hand side. Equality (3.1)is the same as the fundamental identity, but when it is used for the purpose of transforming the left-hand side into the right-hand side, we also call it a transport relation. If we classify v v 1 n−1 transport relations for Z according to whether the actual direction of each of →, ... , → and → is vertical or horizontal, that is, whether each of v , ... ,v (resp. t) is the inﬁnity 1 n−1 (resp. 0) or not, there are 2 diﬀerent patterns of transport relations; see Example 3.1 for instance. 3.1 Examples The settings of k , k , l, z , z and w are as per those at the beginning of this section (the 1 2 1 2 case of n = 2). Example 3.1 (Transport relations for Z )Let t ∈ D and v ∈ B (t). Assume that if k = ∅, 2 1 1 2 1 1 then t = ,and if k = ∅ (resp. k = ∅)and |t|= 1, then t − = 1 (resp. |v | = 1). 1 2 1 v v 1 1 1 1 We deﬁne v ∈ (D \{0}) ∪{∞} by = t − . v v 2 1 (1) The case of v =∞, t = 0: z z w 1 2 Z v ; ( ) t 2 1 k k l 1 2 −→ −→ z z z z 1 2 w 1 2 w =−Z ; v ( ) − Z v ; v . 2 t 2 k k 2 l k 1 k 2 l 1 2 1 2 −→ −→ −→ −→ (2) The case of v =∞, t = 0, k = ∅ (v = 1/t): 1 1 2 z z 1 2 Z ; ( ) t k k l 1 ↑ 2 −→ z z z z 1 2 w 1 2 w = Z ; ( ) − Z ; . 2 1/t t 2 1/t k k l k k l 1 2 1 ↑ 2 −→ −→ −→ (3) The case of v =∞, t = 0(v =−v ): 1 2 1 z z 1 2 Z v ; ( ) 2 1 k k l ↑ 1 2 −→ z z z z 1 2 w 1 2 w =−Z ; ( ) − Z v ; . 2 −v 2 −v k k 1 l k 1 k 1 l 1 2 1 2 −→ −→ −→ (4) The case of v =∞, t = 0, k = ∅ (v =∞): 1 1 2 z z 1 2 Z ; ( ) k k l ↑ 1 ↑ 2 z z z z 1 2 w 1 2 w =−Z ; ( ) + Z ; . 2 2 k k l k k l 1 2 ↑ 1 ↑ 2 ↑ Example 3.2 (Transport relations for duality) We consider a special case of Example 3.1: the case where l = ∅, t = 1. In the case of Example 3.1 (1), we further classify it by whether the actual direction of −→ is vertical or horizontal. Let v ∈ D \{0} satisfying 1 1 Re(v) ≤ . Assume that if k = ∅,thenRe(v) = and if k = ∅, then |v| = 1. Then, we 1 2 2 2 have z z z z 1 2 1 2 Z v ; =−Z ; v , 2 2 k k k k 1 2 1 2 −→ v−1 −→ z z z z 1 2 1 2 Z ; = Z ; (k = ∅), 2 2 2 k k k k 1 2 1 2 ↑ −→ z z z z 1 2 1 2 Z ; = Z ; (k = ∅). 2 2 1 1 k k k k 1 ↑ 2 1 2 −→ 4 Page 12 of 25 Kawamura et al. Res Math Sci (2022) 9:4 These transport relations will be used to reprove the duality relation for MPLs in the next subsection and recover those of Seki–Yamamoto (1.2) by considering the case z = dep(k ) dep(k ) 1 2 ({1} )and z = ({1} ). Example 3.3 (Transport relations for special cases of Z ) In the general case of Z , there 3 3 are 8 patterns of transport relations, considered in the same way as in Example 3.1. Here, we focus on transport relations restricted to the case where it is necessary to calcu- late the multiple connected sum Z (k ; k ; k ). Let k be an index and an element of 3 1 2 3 3 dep(k ) {1, −1} . If the tuple of complex variables to be written onto an index has the form (1, ... , 1), then we omit the tuple from our notation. In this setting, we have (1) The case of (v ,v ,v ) = (1, 1, −1): 1 2 3 Z (k ) ;(k ) ; =−Z k ;(k ) ; − Z (k ) ; k ; . 3 1 → 2 → k 3 1 2 → k −1 3 1 → 2 k −1 3 3 3 −→ −→ (2) The case of (v ,v ,v ) = (1, ∞, ∞), k , k = ∅: 1 2 3 2 3 Z (k ) ;(k ) ; = Z k ;(k ) ; − Z (k ) ; k ; . 3 1 → 2 ↑ k 3 1 2 ↑ k 3 1 → 2 k 3 3 ↑ 3 ↑ (3) The case of (v ,v ,v ) = (∞, 1, ∞), k , k = ∅: 1 2 3 1 3 Z (k ) ;(k ) ; =−Z k ;(k ) ; + Z (k ) ; k ; . 3 1 ↑ 2 → 3 1 2 → 3 1 ↑ 2 k k k 3 3 3 ↑ ↑ (4) The case of (v ,v ,v ) = (∞, ∞, 1), k , k = ∅: 1 2 3 1 2 Z (k ) ;(k ) ; = Z k ;(k ) ; + Z (k ) ; k ; . 1 1 3 1 ↑ 2 ↑ k 3 1 2 ↑ k 3 1 ↑ 2 k 3 3 3 −→ −→ In particular, if all k , k and k are not empty, then we see that Z (k ; k ; k ) belongs to 1 2 3 3 1 2 3 the space of alternating multiple zeta values, which will be deﬁned below, by the recipe explained later in Sect. 5.1. 3.2 Duality for multiple polylogarithms Prior to Sect. 6, we prove the duality for MPLs by using transport relations in Example 3.2, because of its simplicity. Let k be an index and z = (z , ... ,z ) ∈ D , where r:=dep(k). Further, we assume that 1 r all of the following are true: • z , ... ,z ∈ B (1), 1 r 1 •Re(z ) = , •if k is non-admissible, then |z | = 1. We refer this condition as the dual condition. For such a pair , by using transport relations in Example 3.2, we can transport from left to right in the absolutely convergent multivariable connected sum Z as z ∅ ι(z) ∅ z Z ; = ··· = (−1) Z ; , (3.2) 2 2 k k ∅ ∅ where ι(z) = r−(the number of 1’s in z). Here, we deﬁne a pair satisfying the dual condition by this transportation. We also write an explicit deﬁnition. Recall the deﬁnition of the dual index of an admissible index in Sect. 1. Any pair with the dual condition is uniquely expressed as r a −1 r a −1 w , d+1 1 1 w ,..., d d z {1} ,{1} , 1 {1} ,{1} , d {1} = , ,..., a −1 b a −1 b , k l , l , l 1 1 d d 1 {1} , d {1} , d+1 Kawamura et al. Res Math Sci (2022) 9:4 Page 13 of 25 4 where d is a non-negative integer, a , ... ,a , b , ... ,b are positive integers, all w , ... ,w 1 d 1 d 1 d are not 1, l , ... , l are admissible indices, and r :=dep(l ), ... ,r :=dep(l ). Then, 1 d+1 1 1 d+1 d+1 is deﬁned by s w s s d+1 b −1 d b −1 1 {1} , d {1} 1 {1} † d ,..., 1 z {1} , , {1} , , w −1 w −1 = 1 , d ,..., b −1 b −1 k † † † (l ) ,{1} d , a , (l ) {1} , a , (l ) 1 1 d+1 d d r † {1} † † {1} where s :=dep((l ) ), ... ,s :=dep((l ) ). In particular, we have = , 1 1 d+1 d+1 where k is an admissible index and r:=dep(k), s:=dep(k ). The transportation (3.2) with Proposition 2.1 (i) and (iv) gives the following duality relation which was proved by Borwein–Bradley–Broadhurst–Lisonek ˇ in [3, §6.1] using a change of variables for iter- ated integrals; here, we state only the case for absolutely convergent MPLs. dep(k) z Theorem 3.4 Let k be an index and z ∈ D . We assume that the pair satisﬁes the dual condition and write as . Then, we have X ι(z) X Li (z) = (−1) Li (z ). Here, we recall the deﬁnition of the alternating multiple zeta value. For a non-empty index k = (k , ... ,k )and = ( , ... , ) ∈{1, −1} satisfying (k , ) = (1, 1), we deﬁne 1 r 1 r r r ζ (k; )by m m 1 r ··· ζ (k; ):= . k k 1 r m ··· m 0<m <···<m 1 1 r We use bar notation for indicating signs; e.g., ζ (1, 2, 3, 4) = ζ (1, 2, 3, 4; −1, 1, −1, 1). Example 3.5 The identity ζ (1, 2) =−Li 1, 2,1 is deduced from −1,1 ∅ −1,1 ∅, 1 −1 1 1, Z ; = Z ; = Z ; =−Z . 2 2 2 2 1,2 ∅ 1,1 2 ∅, 1 1 2, 1 Borwein, Bradley, Broadhurst and Lisonek ˇ remarked that this is “a result that would doubtless be diﬃcult to prove by naïve series manipulations alone” in [3, Example 6.1]. However, our proof is contrary to their prediction; if we state the proof of Theorem 2.4 in this case, then it is quite simple. Example 3.6 Let r be a positive integer. Take (z , ... ,z ) ∈ (B (1) \{∞}) satisfying 1 r 1 Re(z ) = and |z | = 1. Then, the identity 1 r z z r 1 X r X Li r (z , ... ,z ) = (−1) Li r , ... , 1 r {1} {1} z − 1 z − 1 r 1 is deduced from zr r 1 z ,...,z ,..., ,..., 1 r ∅ z z r ∅ 1 r−1 z −1 z −1 z −1 r r Z ; =−Z ; = ··· = (−1) Z ; 1 . 2 2 ,..., 2 1,...,1 ∅ 1 1 ∅ 1,...,1 4 Boundary condition We call the following fact the boundary condition for multivariable connected sums: every absolutely convergent Z -value is written explicitly as a Z-linear combination of absolutely convergent shuﬄe-type MPLs. We explain this fact in detail. 4 Page 14 of 25 Kawamura et al. Res Math Sci (2022) 9:4 Let r be a positive integer, k = (k , ... ,k )anindex and ξ , ... , ξ complex numbers 1 r 1 r satisfying ξ ≤ 1 for all 1 ≤ j ≤ r and that if k = 1, then |ξ | < 1. Then, the i r r i=j harmonic-type multiple polylogarithm Li (ξ , ... , ξ )isdeﬁnedby 1 r m m 1 r ξ ··· ξ ∗ 1 Li (ξ , ... , ξ ):= . 1 r k k 1 r m ··· m 0<m <···<m 1 1 r Let z = (z , ... ,z ) ∈ (D \{0}) satisfying that if k is non-admissible, then |z | < 1. By 1 r r ∗ X deﬁnition, there are the following relationships between Li and Li : k k z z 1 r−1 X ∗ Li (z , ... ,z ) = Li , ... , ,z , (4.1) 1 r r k k z z 2 r r r ∗ X Li (ξ , ... , ξ ) = Li ξ , ξ , ... , ξ ξ , ξ . (4.2) 1 r i i r−1 r r k k i=1 i=2 Furthermore, let s be a positive integer, l = (l , ... ,l )anindex and w = (w , ... ,w ) ∈ 1 s 1 s (D \{0}) . We replace the assumption about z with the following: if k = l = 1, then r r s |z | < 1or |w | < 1. By deﬁnition, we have r s ⎛ ⎞ m m 1 r−1 z z 1 r−1 ··· z z ⎜ 2 r ⎟ z w Z = 1 ⎝ ⎠ k l 1 r−1 m ··· m n=1 0<m <···<m <n 1 r−1 1 r−1 ⎛ ⎞ q q 1 s−1 w w 1 s−1 ··· n w w (z w ) 2 s ⎜ ⎟ r s × . ⎝ ⎠ l l k +l −1 1 s−1 r s q ··· q 1≤q ≤···≤q ≤n 1 s−1 1 s−1 Since the product of truncated sums for a ﬁxed n obeys the harmonic product rule, z w Z is writtenasa Z-linear combination of harmonic-type MPLs, and hence of k l shuﬄe-type by (4.2). For example, we have z ,z w ,w 1 2 1 2 1 1,2 1,1 ⎛ ⎞ ⎛ ⎞ m q n 1 1 z w w 1 1 1 z w w (z w ) ⎜ 2 ⎟ ⎜ 2 2 ⎟ 2 2 = + ⎝ ⎠ ⎝ ⎠ m q n n 1 1 n=1 0<m <n 0<q <n 1 1 z w w z 1 1 1 1 ∗ ∗ = Li , ,z w + Li , ,z w 2 2 2 2 1,1,2 1,1,2 z w w z 2 2 2 2 z w z 1 1 1 ∗ ∗ + Li ,z w + Li ,z w 2 2 2 1 2,2 1,3 z w z 2 2 2 X X = Li (z w ,z w ,z w ) + Li (z w ,z w ,z w ) 1 1 2 1 2 2 1 1 1 2 2 2 1,1,2 1,1,2 X X + Li (z w ,z w ) + Li (z w ,z w ). 1 1 2 2 1 1 2 1 2,2 1,3 Remark 4.1 If z = ··· = z = w = ··· = w = 1, then another decomposition 1 r 1 s approach using 2-posets and associated integrals is attributable to Kaneko and Yamamoto; see [6,§4]. Kawamura et al. Res Math Sci (2022) 9:4 Page 15 of 25 4 5 Evaluations of the multivariable connected sum 5.1 Recipe for evaluations First, we introduce the notion of transportable variables. This is just a condition to com- plete all transportation process while maintaining absolute convergence of multivariable connected sums. We use the notation [n]:={1, ... ,n} for a positive integer n. Deﬁnition 5.1 Let r , ... ,r and s be positive integers, where n is an integer at least 2. 1 n (1) (1) (n) (n) Let z = (z , ... ,z ), ... , z = (z , ... ,z )and w = (w , ... ,w ) be elements of 1 r n r 1 s 1 1 1 n r r s 1 n D , ... , D and D , respectively. We declare that (z , ... , z ; w)is transportable if there 1 n exists j ∈ [n] such that for any non-empty subset J ⊂ [n] \{j}, the following holds true: • For every (a ) (a ∈ [r ]) and every component w of w, i i∈J i i k (i) (z ) ∈ B (w ). i∈J #J k • For every component w of w with |w |= 1, k k w − = 1 for all i ∈ [n] \{j}. (i) r s If every z is an element of {1, −1} and w ∈{0, 1, −1} ,then(z , ... , z ; w)istransportable i 1 n by deﬁnition. Given the above, we can provide an explicit decomposition of the multivariable con- nected sum according to the following recipe: Theorem 5.2 Let r , ... ,r and s be positive integers, where n is an integer at least 2. 1 n Let k , ... , k and l be indices satisfying dep(k ) = r , ... , dep(k ) = r , dep(l) = s. Let 1 n 1 1 n n r r s s 1 n z , ... , z and w be elements of D , ... , D and D , respectively. We assume that if l ={1} 1 n (resp. l ={1} ),then (z , ... , z ; w)(resp. (z , ... , z ;(w, 0))) is transportable. Then, the 1 n 1 n z z 1 n multivariable connected sum Z ; ... ; is expressed as a Z-linear combination of k k l 1 n absolutely convergent multiple polylogarithms. Explicitly, we obtain such an expression by the following procedures: (i) Take an integer j ∈ [n] in Deﬁnition 5.1 and replace the jth component with the nth component by Proposition 2.1 (iii). z z 1 n w (ii) Apply the transport relation (3.1) repeatedly until Z ; ... ; is expressed as a k k l 1 n Z-linear combination of Z -values by Proposition 2.1 (iv). n−1 (iii) If n ≥ 3, then apply the transport relation (3.1)(replace n with n − 1) to each Z - n−1 value repeatedly until it is expressed as a Z-linear combination of Z -values. n−2 z z 1 n w (iv) Repeat the same procedure as in (iii) to express Z ; ... ; as a Z-linear com- k k l 1 n bination of Z -values. (v) Apply the boundary condition in Sect. 4 to each Z -value obtained in (iv); then, we have the desired Z-linear combination of absolutely convergent multiple polylogarithms. Proof All that needs to be checked is that suﬃcient conditions for applying the transport relation in (ii)–(iv) are satisﬁed, which is accomplished by the assumption and Deﬁni- tion 5.1. Proof of Theorem 1.1 Let k be a positive integer and k , ... , k , l indices as in the state- 1 n ment. We decompose Z (k ; ... ; k | l)asa Z-linear combination of Z -values according n 1 n 1 to the recipe (ii)–(iv) in Theorem 5.2; the transportable condition is satisﬁed for all j ∈ [n] 4 Page 16 of 25 Kawamura et al. Res Math Sci (2022) 9:4 in this case, and thus, we may assume that j = n. Then, we see that each of the Z -value belongs to ⎧ ⎫ r, s ∈ Z ,r + s ≤ k − 1, ⎪ ⎪ >0 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ s k ∈ I , k ∈ I with wt(k) + wt(k ) = k, r s 1,z ,...,z {1} 2 r Z , k k ⎪ ⎪ k or k is admissible, and ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 1 1 z , ... ,z ∈{1, −1, − , ... , − } 2 r 2 n−1 where I (resp. I ) denotes the set of indices whose depth is r (resp. s). By the boundary r s condition in Sect. 4, it is straightforward to see that every Z -value belonging to the above set is expressed as a Z-linear combination of elements of 1≤r≤k−2,k is admissible with wt(k)=k−1, Li (1,z , ... ,z ) . 1 1 2 r dep(k)=r,and z ,...,z ∈{1,−1,− ,...,− } 2 r 2 n−1 Finally, every element of this set coincides with an element of MPL(n, k)uptosignbythe duality (Theorem 3.4). 5.2 Examples Example 5.3 (= Equality (1.4)) Z (1; 1 | 1, 1) = 3ζ (3). Proof By the transport relation, we have Z (1; 1 | 1, 1) = Z (∅;2 | 1, 1) + Z (1; 2) = Z (2 | 1, 1) + Z (3). 2 2 2 1 1 Boundary conditions for this case are Z (2 | 1, 1) = ζ (1, 2) + ζ (3) = 2ζ (3) and Z (3) = 1 1 ζ (3). Example 5.4 (= Equality (1.7)) Z (1; 1 | 2, 1) = ζ (4). Proof By the transport relation, we have Z (1; 1 | 2, 1) = Z (∅;2 | 2, 1) + Z (1; 2 | 2) 2 2 2 ∅ 1,−1 1,−1 1 1 = Z (2 | 2, 1) − Z ; − Z ; 1 2 2 ∅ 2,1 2 1 2,1 1,−1 1,−1 = Z (2 | 2, 1) − Z − Z . 1 1 1 2,1 2 2,2 Boundary conditions for this case are 1,−1 1,−1 Z (2 | 2, 1) = ζ (2, 2) + ζ (4),Z = ζ (2, 2),Z = ζ (2, 2). 1 1 1 2,1 2 2,2 Hence, we have Z (1; 1 | 2, 1) = ζ (2, 2) + ζ (4) − 2ζ (2, 2). 3 3 The facts that ζ (2, 2) = ζ (4) and ζ (2, 2) =− ζ (4) are well-known. 4 16 Example 5.5 (= Equality (1.5)) For a positive integer n,wehave 1 1 Z (1; 1; ... ;1) = n!Li 1, , ... , . n+1 n−1 {1} ,2 2 n Kawamura et al. Res Math Sci (2022) 9:4 Page 17 of 25 4 Proof By the transport relation (3.1) and Proposition 2.1 (iv), z z 1 1 1 1 Z ; ... ; ; =−rZ ; ... ; ; r+1 r 1 k k 1 1 1 1 1−r −→ dep(k) holds for an integer r ≥ 2, a non-empty index k and z ∈ D . Applying this repeatedly and then applying the transport relation for the duality, we have 1 1 ,..., n−1 1,− − , 1 −1 n−1 2 Z (1; 1; ... ;1) = (−1) n!Z ; n+1 2 ,..., 1 1, 1 1, 1 1 n−1 X = (−1) n!Li 1, − , ... , − , −1 . n−1 {1} ,2 n − 1 2 Since 1 1 1 1 X n−1 X Li 1, − , ... , − , −1 = (−1) Li 1, , ... , (5.1) n−1 n−1 {1} ,2 {1} ,2 n − 1 2 2 n holds as a consequence of the duality (Theorem 3.4), the proof is completed. Example 5.6 (Equality (1.1) with variables) Let z and z be complex numbers whose 1 2 absolute values are less than 1. Assume that Re(z ) < holds for i = 1 or 2. Then, we have z z 1 2 Z ; = Li (z ) + Li (z ) − Li (z + z − z z ). 2 2 1 2 2 2 1 2 1 2 1 1 Here, Li (z) is the dilogarithm. Proof We may assume that z = 0and Re(z ) < . By the transport relation and Propo- 1 1 sition 2.1 (iv),(i),wehave 1 z z , 1 z z ∅ 2 X 1 2 z −1 Z ; =−Z ; 1 =−Li z , . (5.2) 2 2 2 1, 1 1 ∅ 1,1 z − 1 By substituting z (z − 1)/z (resp. z /(z − 1)) into x (resp. y) in Zagier’s formula [15, 2 1 1 1 1 Proposition 1] xy − y y Li (x, y) = Li − Li − Li (xy)(|xy| < 1, |y| < 1), 2 2 2 1,1 1 − y y − 1 we have the desired formula. 6 Relations among special values of multiple polylogarithms 6.1 Recipe for relations We have already derived the duality for MPLs by using our transport relations in Sect. 3.2. In this subsection, we describe a general recipe for obtaining relations among MPLs based on transport relations and evaluations of multivariable connected sums (Theorem 5.2). We consider data • n:anintegeratleast 2, • k , ... , k and l: non-empty indices, 1 n−1 (1) (1) (n−1) (n−1) r r 1 n−1 • z = (z , ... ,z ) ∈ D , ... , z = (z , ... ,z ) ∈ D and w = 1 r n−1 r 1 1 1 n−1 (w , ... ,w ) ∈ D , where r = dep(k ), ... ,r = dep(k )and s = dep(l) 1 s 1 1 n−1 n−1 satisfying the following assumptions: 4 Page 18 of 25 Kawamura et al. Res Math Sci (2022) 9:4 •If l is non-admissible, then = w , (i) 1≤i≤n−1 i k is non-admissible and if l is admissible, then = 0. (i) 1≤i≤n−1 i k is non-admissible • For every non-empty subset J ⊂ [n − 1], every (a ) (a ∈ [r ]) and every component i i∈J i i w of w, (i) (z ) ∈ B (w ). i∈J #J k •If l ={1} , then the second assumption with w replaced by (w, 0) is valid. • For every component w of w with |w |= 1, k k w − = 1 for all i ∈ [n − 1]. (i) (1) •If n = 2 and both k and l are non-admissible, then |z | < 1or |w | < 1holds. 1 r s For such given data, we can obtain a relation among MPLs by the following procedure: (i) By Proposition 2.1 (iv), we have z z z n−1 z n−1 1 ∅ w 1 w Z ; ... ; ; = Z ; ... ; (6.1) n n−1 k k ∅ l k k l 1 n−1 1 n−1 and then we can decompose the right-hand side as a Z-linear combination of MPLs by Theorem 5.2 or the boundary condition. (ii) On the other hand, by using the transport relation (3.1) and Propositions 2.1 (iv) as needed, we can decompose the left-hand side of (6.1)asa Z-linear combination of Z -values and Z -values. Then, we decompose all these Z -values and Z -values n n−1 n n−1 as Z-linear combinations of MPLs by Theorem 5.2 or the boundary condition. (iii) By comparing the output results of (i) and (ii), we obtain a relation among MPLs. Using assumptions regarding the data and contents in Sects. 3–5, it is true that the above procedure yields a relation among MPLs; note that the relation can be tautological as in (1.3). The family of relations obtained by executing the above procedure on data satisfying s s n = 2, l = ({1} )and w = ({1} ) is actually a natural extension of the family of Ohno’s relations for MZVs to the MPLs case. We prove Ohno’s relation for MPLs in the next subsection; see Theorem 6.10. The above description is the concrete content of Theorem 1.4. 6.2 Ohno’s relation for multiple polylogarithms For a non-negative integer h, we use the following shorthand notation for multivariable connected sums: h+1 z z z z (h) 1 2 1 2 {1} Z ; :=Z ; . k k k k h+1 1 2 1 2 {1} dep(k ) dep(k ) 1 2 Let k , k be indices, z ∈ D , z ∈ D and v ∈ B:=B (1) \{∞}. We assume 1 2 1 2 1 that if k = ∅ (resp. k = ∅), then Re(v) = (resp. |v| = 1). Then, by transport relations 1 2 2 Kawamura et al. Res Math Sci (2022) 9:4 Page 19 of 25 4 for Z (Example 3.1 (1), (2)), the following relations hold: (h) z z (h) z z (h−1) z z 1 2 1 2 1 2 Z v ; =−Z ; v − Z v ; v (v = 1), k k k k k k 1 2 1 2 1 2 −→ v−1 −→ v−1 −→ −→ z z z z z z (h) 1 2 (h) 1 2 (h−1) 1 2 Z ; = Z ; + Z ; (k = ∅), 1 1 k k k k k k 1 2 1 2 ↑ 1 2 ↑ −→ −→ z z z z z z (h) 1 2 (h) 1 2 (h−1) 1 2 Z ; = Z ; − Z ; (k = ∅). 1 1 1 k k k k k k 1 2 1 2 1 2 ↑ ↑ −→ −→ (−1) Here, set Z (∗; ∗):=0; the case where h = 0 in the above relations gives the transport relations for the duality (Example 3.2). It can be proved by direct calculation that the space of relations obtained by restricting (h) to the use of transport relations for {Z } coincides with the space of Ohno’s relations h≥0 for MPLs. However, in this subsection, we derive Ohno’s relations for MPLs more clearly in terms of a formal power series ring over a Hoﬀman-type algebra. This also gives a new proof of Ohno’s relation for MZVs; compare this with another proof using connected sums by the third author and Yamamoto [12]. First, we prepare some terminology of the Hoﬀman-type algebra appropriately. Deﬁnition 6.1 Prepare letters x and e for all z ∈ B.Let A be a non-commutative polynomial algebra A:=Qx, e (z ∈ B). Furthermore, we deﬁne its subalgebras A and A by A :=Q ⊕ e A ⊃ z∈B ( ( ( A :=Q ⊕ Qe ⊕ e Ax ⊕ e Ae . z z z 1 1 z∈B, z∈B, Re(z)= z∈B, Re(z)= 2 2 Re(z)= ,|z|=1 z ∈B, |z |=1 1 z k −1 k −1 1 r We say that a word w ∈ A is associated with when w = e x ··· e x , where z z k 1 r z = (z , ... ,z )(z ∈ B)and k = (k , ... ,k )(k ∈ Z ). In addition, 1 ∈ A is associated 1 r i 1 r i >0 ∅ 0 with . Note that a word of A is associated with a pair satisfying the dual condition in Sect. 3.2. We deﬁne a Q-linear map L: A [[t]] → C[[t]] by ∞ ∞ i i L w t := L(w )t i i i=0 i=0 and a Q-linear map L: A [[t]] → C[[t]] by ∞ ∞ i i L w t := L(w )t . i i i=0 i=0 Here, X (h) z ∅ h L(w):=Li (z), L(w):= Z ; t k ∅ h=0 0 z for each word w ∈ A associated with . k 4 Page 20 of 25 Kawamura et al. Res Math Sci (2022) 9:4 Deﬁnition 6.2 Let A[[t]] be the formal power series ring over A. We deﬁne two auto- morphisms σ , ρ and two anti-automorphisms τ, τ on A[[t]] by −1 σ (x) = x, σ (e ) = e (1 − xt) (z ∈ B), z z −1 ρ(x) = x, ρ(e ) = e (1 − e t) (z ∈ B), z z z −1 z z τ(x) = e , τ(e ) = x, τ(e ) =−e (1 − e t) (z ∈ B \{1}), 1 1 z z−1 z−1 −1 −1 −1 τ (x) = e (1 + e t) , τ (e ) = x(1 − xt) , τ (e ) =−e z (1 + e z t) (z ∈ B \{1}) 1 1 1 z z−1 z−1 −1 −1 and σ (t) = ρ(t) = τ(t) = τ (t) = t. Images of the generators under σ and ρ are given by −1 −1 σ (x) = x, σ (e ) = e (1 − xt)(z ∈ B), z z −1 −1 −1 ρ (x) = x, ρ (e ) = e (1 + e t) (z ∈ B), z z z −1 −1 −1 −1 with σ (t) = ρ (t) = t. Furthermore, τ = τ and τ = τ hold. Lemma 6.3 ρ ◦ τ = τ ◦ ρ. Proof It is suﬃcient to check that (ρ ◦ τ )(u) = (τ ◦ ρ)(u) holds for u ∈{x}∪{e | z ∈ B}. Proposition 6.4 (Boundary condition) For every w ∈ A [[t]],wehave L(w) = L ((σ ◦ ρ)(w)) . k −1 k −1 0 1 r Proof It is suﬃcient to show the case w = e x ··· e x ∈ A . By deﬁnition, we z z 1 r have −1 k −1 −1 k −1 1 r (σ ◦ ρ)(w) = σ e (1 − e t) x ··· e (1 − e t) x z z z z 1 1 r r −1 k −1 −1 k −1 1 r = e (1 − xt − e t) x ··· e (1 − xt − e t) x z z z z 1 1 r r f k −1 f k −1 h 1 1 r r = e (e + x) x ··· e (e + x) x t . z z z z 1 1 r r h=0 f ,...,f ≥0 1 r f +···+f =h 1 r Since f k −1 f k −1 1 1 r r L e (e + x) x ··· e (e + x) x z z z z 1 1 r r f ,...,f ≥0 1 r f +···+f =h 1 r m −m m m −m r r−1 1 2 1 z z ··· z r z ,...,z 1 2 (h) 1 r ∅ = Z ; k ,...,k ∅ 1 r k k 1 r m ··· m i ··· i r 1 0<m <···<m 1 1 r 0=i <i ≤···≤i ≤m 0 1 h r holds, we have L ((σ ◦ ρ)(w)) = L(w) by deﬁnition. (h) Deﬁnition 6.5 For a non-negative integer h,wedeﬁne a Q-bilinear map Z : A × A → C by (h) (h) z z 1 2 Z (w ; w ):=Z ; , 1 2 k k 1 2 Kawamura et al. Res Math Sci (2022) 9:4 Page 21 of 25 4 z z 1 2 where A := e A (non-unital) and w and w are associated with and , z 1 2 z∈B k k 1 2 respectively. Furthermore, we deﬁne a Q-bilinear map Z: A [[t]] × A [[t]] → C[[t]] by ⎛ ⎞ ∞ ∞ ∞ ∞ i j i+j ⎝ ⎠ Z w t ; w t := Z(w ; w )t i i j j i=0 j=0 i=0 j=0 and (h) h Z(w ; w ):= Z (w ; w )t . i i j j h=0 Proposition 6.6 (Transport relation) We have the following: (i) For w ,w ∈ A [[t]] and u ∈ A[[t]], 1 2 Z(w u; w ) = Z(w ; w τ (u)). 1 2 1 2 (ii) For w ∈ A [[t]] and u ∈{x}∪{e | z ∈ B, |z| = 1}, L(wu) = Z(w; τ (u)). (iii) For w ∈ A [[t]] and u ∈{e | z ∈ B, Re(z) = }, Z(u; w) = L(wτ (u)). (iv) For z ∈ B with Re(z) = and |z| = 1, L(e ) = L(τ (e )). z z Proof It is suﬃcient to show the case where w ,w ,w are words and u is a letter. In this 1 2 (h) case, it follows immediately from the transport relations for {Z } . h≥0 Theorem 6.7 For w ∈ A [[t]],wehave L(σ (w)) = L(σ (τ(w))). Proof By Proposition 6.6,wehave L(w) = L(τ (w)). Therefore, by Proposition 6.4 and Lemma 6.3,wehave L σ (ρ(w)) = L σ ((ρ ◦ τ )(w)) = L (σ ◦ τ)(ρ(w)) . ( ) ( ) −1 0 0 Since ρ is invertible and ρ (A [[t]]) ⊂ A [[t]], we have L(σ (w)) = L(σ (τ(w))) for all w ∈ A [[t]]. To restate Theorem 6.7 without terminology of the Hoﬀman-type algebra, we deﬁne some new notation. Recall the notion of the dual condition, and ι(z) from Sect. 3.2. Deﬁnition 6.8 Let r be a positive integer, k = (k , ... ,k )anindex and z = (z , ... ,z ) ∈ 1 r 1 r r z D satisfying that if k = 1, then |z | = 1. For a non-negative integer h,wedeﬁne O r r h by O := Li (z). k +c ,...,k +c 1 1 r r c ,...,c ≥0 1 r c +···+c =h 1 r 4 Page 22 of 25 Kawamura et al. Res Math Sci (2022) 9:4 dep(k) z Deﬁnition 6.9 Let k be a non-empty index and z ∈ D .Set d = ι(z). When is written as a a ,..., z , d+1 1 z d z {1} , 1 {1} , d {1} = , ,..., k l l , k , 1 k , k 1 d d d+1 where a , ... ,a are non-negative integers, z , ... ,z ∈ D \{1}, k , ... , k are 1 1 1 d+1 d d+1 indices with dep(k ) = a , ... , dep(k ) = a and l , ... ,l are positive integers. 1 1 d+1 d+1 1 d Let b , ... ,b be non-negative integers. Then, we deﬁne by 1 d b ,...,b 1 d a b +1 a b +1 1 ,..., d d+1 1 {z } {1} d , {z } , {1} z {1} , 1 := . ,..., k b b b ,...,b k , k , k 1 1 1 d d {1} ,l d {1} ,l , d+1 1 d Theorem 6.10 (Ohn o’s relation for multiple polylogarithms) Let k be a non-empty index dep(k) z z and z ∈ D . We assume that the pair satisﬁes the dual condition and write as k k .Set d = ι(z). Then, for any non-negative integer h, we have z z O = (−1) O . h h−i b ,...,b i=0 b ,...,b ≥0 b +···+b =i In particular, when d = 0(then k is admissible),thismeans dep(k) dep(k ) {1} {1} O = O , h h which is the well-known Ohno’s relation for multiple zeta values [8]. h 0 Proof In Theorem 6.7, we compare the coeﬃcients of t for the case where w ∈ A is a word. Then, we have the desired formula from various deﬁnitions of notation. ∗ dep(k)−1 X dep(k) In the following corollary, we use the symbol Li (z):=Li ({1} ,z) = Li ({z} ) k k for the 1-variable multiple polylogarithm. Corollary 6.11 (The depth 1 case of the Landen connection formula,[9]) For a positive integer k and a complex number z satisfying |z| < 1 and Re(z) < ,wehave Li (z) =− Li , k k z − 1 k with wt(k)=k where k runs over all indices whose weight equal k. z z Proof This is the case where = ( ) and h = k − 1inTheorem 6.10. 6.3 Other relations Example 6.12 For k ≥ 2, we have k−1 k−2 k−j+1 k−j ζ (k) = (−1) ζ ({1} , 2) + (−1) ζ ({1} , j). (6.2) j=2 k−1 k−1 Proof Consider the data n = 2, k = ({1} ), z = ({1} ), l = (2) and w = (1) in the 1 1 recipe in Sect. 6.1. First, with the duality for MZVs, we have k−1 {1} ∅ k−2 Z ; = ζ ({1} , 2) = ζ (k). k−1 ∅ 2 {1} Kawamura et al. Res Math Sci (2022) 9:4 Page 23 of 25 4 On the other hand, by Example 3.1 (3), k−1 {1} Z ; k−1 ∅ {1} k−2 k−1 {1} {1} −1 1 −1 =−Z ; − Z ; 2 2 k−2 2 k−1 1 1 {1} {1} = ··· k−1 j−1 k−j+1 {−1} {1} {−1} k−1 ∅ k−j+1 = (−1) Z ; + (−1) Z ; 2 2 ∅ k−1 j−1 k−j+1 {1} {1} {1} j=2 holds, and by the transport relation for the duality (Example 3.2), j−1 k−j+1 {1} {−1} X k−j+1 k−j Z ; = ··· = Li ({−1} ) = ζ ({1} , j) j−1 k−j+1 k−j {1} ,j {1} {1} holds for each j. Furthermore, k−1 {−1} ∅ X k−1 k−2 Z ; = Li ({−1} ) = ζ ({1} , 2) ∅ k−1 k−2 {1} ,2 {1} holds and thus, we have the desired formula. 1−k Remark 6.13 By using ζ (k) =−(1 − 2 )ζ (k), equality (6.2)isrewrittenas ⎛ ⎞ k−1 k−1 k−2 j k−j ⎝ ⎠ ζ (k) = (−2) 2ζ ({1} , 2) + (−1) ζ ({1} , j) , j=3 which was recently proved by Dilcher–Vignat [4, Theorem 4.1]; their proof is diﬀerent from ours. The case where k = 3 is the famous formula ζ (3) = 8ζ (1, 2); see [2]. Example 6.14 Let z ,z ,z be elements of {z ∈ C | 0 < |z| < 1, Re(z) < } satisfying 1 2 3 1 1 1 + + = 1. Then, we have z z z 1 2 3 z z z 2 3 1 X X X Li z , + Li z , + Li z , = 0. 1 2 3 1,1 1,1 1,1 z − 1 z − 1 z − 1 2 3 1 Proof By the fundamental identity, we have z z ∅ z ∅ z ∅ z z 1 2 1 3 2 3 Z ; ; + Z ; ; + Z ; ; = 0 3 3 3 1 1 ∅ 1 ∅ 1 ∅ 1 1 and by Propositions 2.1 (iv), this is z z z z z z 1 2 1 3 2 3 Z ; + Z ; + Z ; = 0. (6.3) 2 2 2 1 1 1 1 1 1 Then, we have the desired formula by a similar calculation as in equality (5.2). Remark 6.15 By applying Example 5.6 to (6.3), we have the six-term relation for diloga- rithms (z ,z ,z as in Example 6.14) 1 2 3 1 z z z z z z 1 2 2 3 3 1 Li (z ) + Li (z ) + Li (z ) = Li − + Li − + Li − 2 1 2 2 2 3 2 2 2 2 z z z 3 1 2 attributable to Kummer and Newman; see [15,p.9]. Example 6.16 Let z ,z ,z ,z be elements of {z ∈ C | 0 < |z| < 1, Re(z) < } satisfying 1 2 3 4 1 1 1 1 + + + = 1and |g(a, b)|≤ 1 for (a, b) = (z ,z ), (z ,z ), (z ,z ), (z ,z ), where 1 2 2 3 3 4 4 1 z z z z 1 2 3 4 ab g(a, b):= . Then, we have an eight-term relation for Li , 1,1,1 ab−a−b L(z ,z ,z ) + L(z ,z ,z ) + L(z ,z ,z ) + L(z ,z ,z ) = 0, 1 2 3 2 3 4 3 4 1 4 1 2 4 Page 24 of 25 Kawamura et al. Res Math Sci (2022) 9:4 where a b X X L(a, b, c):=Li c, g(a, b), + Li c, g(a, b), . 1,1,1 1,1,1 a − 1 b − 1 Proof By the fundamental identity, we have z z z ∅ z z ∅ z z ∅ z z ∅ z z z 1 2 3 1 2 4 1 3 4 2 3 4 Z ; ; ; + Z ; ; ; + Z ; ; ; + Z ; ; ; = 0 4 4 4 4 ∅ ∅ ∅ ∅ 1 1 1 1 1 1 1 1 1 1 1 1 and by Proposition 2.1 (iv), this is z z z z z z z z z z z z 1 2 3 1 2 4 1 3 4 2 3 4 Z ; ; + Z ; ; + Z ; ; + Z ; ; = 0. 3 3 3 3 1 1 1 1 1 1 1 1 1 1 1 1 According to the recipe for evaluations (Theorem 5.2), we have z z z z z , z z , 1 2 3 ∅ 2 3 g(z ,z ) 1 ∅ 3 g(z ,z ) 1 2 1 2 Z ; ; =−Z ; ; − Z ; ; 3 3 3 1 1 1 ∅ 1 1, 1 ∅ 1, 1 1 z z 2 1 X X = Li z ,g(z ,z ), + Li z ,g(z ,z ), 3 1 2 3 1 2 1,1,1 1,1,1 z − 1 z − 1 2 1 = L(z ,z ,z ). 1 2 3 The remaining three Z -values can be calculated in the same way. Acknowledgements The authors would like to express their sincere gratitude to Professor Masanobu Kaneko and Professor Tetsushi Ito for their helpful comments. The ﬁrst author would like to thank Dr. Junpei Tsuji for guiding him through the program code for experiments on our proof of Ohno’s relation. The authors are grateful to the anonymous referee for valuable comments and useful suggestions to improve the manuscript. Author details Department of Mathematics, Faculty of Science Division I, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan, Department of Information and Computer Science, Kanazawa Institute of Technology, 7-1 Ohgigaoka, Nonoichi-shi, Ishikawa 921-8501, Japan, Department of Mathematical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara-shi, Kanagawa 252-5258, Japan. Received: 13 March 2021 Accepted: 11 October 2021 References 1. Accélération pour zeta(2), forum post on Les-Mathematiques.net. http://www.les-mathematiques.net/phorum/read. php?4,837804,843195 2. Borwein, J.M., Bradley, D.M.: Thirty-two Goldbach variations. Int. J. Number Theory 2, 65–103 (2006) 3. 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World Scientiﬁc, Hackensack (2016) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.
Research in the Mathematical Sciences – Springer Journals
Published: Mar 1, 2022
Keywords: Multiple zeta values; Multiple polylogarithms; Connected sums; Connector; Ohno’s relation
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