Alternating Multiple T -Values: Weighted Sums, Duality, and Dimension Conjecture
aa r X i v : . [ m a t h . N T ] S e p Alternating Multiple T -Values: Weighted Sums,Duality, and Dimension Conjecture Ce Xu a, ∗ Jianqiang Zhao b, † a. School of Mathematics and Statistics, Anhui Normal University,Wuhu 241000, P.R. Chinab. Department of Mathematics, The Bishop’s School, La Jolla,CA 92037, United States of America Dedicated to professor Masanobu Kaneko on the occasion of his 60th birthday
Abstract
In this paper, we define some weighted sums of the alternating multiple T -values(AMTVs), and study several duality formulas for them by using the tools developed in ourprevious papers. Then we introduce the alternating version of the convoluted T -values andKaneko-Tsumura ψ -function, which are proved to be closely related to the AMTVs. At the endof the paper, we study the Q -vector space generated by the AMTVs of any fixed weight w andprovide some evidence for the conjecture that their dimensions { d w } w ≥ form the tribonaccisequence 1, 2, 4, 7, 13, .... Keywords : multiple zeta values, Kaneko–Tsumura multiple T -values, alternating multiple T -values, weight sum formulas, duality, tribonacci sequence. AMS Subject Classifications (2020):
Recently, several variants of classical multiple zeta values of level 2 called multiple t -values (abbr. MtVs), multiple T -values (abbr. MTVs) and multiple mixed values (abbr. MMVs) wereintroduced and studied in Hoffman [13], Kaneko–Tsumura [16,17] and Xu–Zhao [27], defined by t ( k , . . . , k r ) := X
1. We call k + · · · + k r and r the weight and depth ,respectively.The systematic study of MZVs began in the early 1990s with the works of Hoffman [11]and Zagier [29]. Due to their surprising and sometimes mysterious appearance in the study ofmany branches of mathematics and theoretical physics, these special values have attracted a lotof attention and interest in the past three decades (for example, see the book by the secondauthor [32]). Clearly, MtVs and MTVs are special cases of MMVs. Moreover, it is obvious thatMtVs satisfy the series stuffle relation, however, it is highly nontrivial to see that MTVs can beexpressed using iterated integral and satisfy the integral shuffle relation (see [17, Theorem 2.1]).Further, in [27], we found that MMVs satisfy the series stuffle relations and integral shufflerelations.The subject of this paper is the alternating MTVs (abbr. AMTVs) which are natural leveltwo generalizations of MTVs. We shall derive a few weight sum formulas and a duality formula(see Theorem 2.2.1). The weight sum formula of MZVs is widely known and its variants areenormous (see, for e.g., [7–9, 12, 14, 19, 21] and [32, Ch. 5]). For Hoffman’s MtVs, a number ofmathematicians also studied their weighed sum formulas in [15, 22, 31].In contrast, we know very little about the weighed sum formulas for MTVs. For these values,certain formulas are found only in depths 2 and 3. For details, see Kaneko and Tsumura [17,Thms. 3.2 and 3.3]. Recently, the first author has considered the following weighted sums ofAMTVs W ( k, r ) := X k ··· + kr = k,k ,...,kr ≥ T ( k , . . . , k r − , k r + m − , (1.1)and proved two duality formulas for the weighted sums W ( k, r ) (see [26, Thms. 1.1 and 1.2]).Here T ( k , . . . , k r − , k r ) stands for a kind of AMTV, which is defined for positive integers k , . . . , k r by T ( k , . . . , k r − , k r ) := 2 r X 4) = T (2 , , , − , , − , . If one or more of the σ j ’s is − alternating multiple T -value (abbr. AMTV). In fact, for any k = ( k , k , . . . , k r ) ∈ N r , ε = ( ε , . . . , ε r ) ∈ {± } r , and σ = ( σ , . . . , σ r ) ∈ {± } r with ( k r , σ r ) = (1 , M σ ( k ; ε ) := X 1. For brevity, we put a check on top of the component k j if ε j = − 1. For example, M σ ,σ ,σ ( k , k , ˇ k ) = 8 X <ℓ 1) (see [32, Ch. 15]), in which case we call ( k , η ) admissible .The level two colored MZVs are called alternating MZVs. In this case, namely, when( η , . . . , η r ) ∈ {± } r and ( k r , η r ) = (1 , ζ ( k ; η ) = Li k ( η ). Further, we put a bar on topof k j if η j = − 1. For example, ζ (¯2 , , ¯1 , 4) = ζ (2 , , , − , , − , . It is clear that every non-alternating MMV (MtV or MTV) can be written as a Q -linearcombination of alternating MZVs. Further, (1.2) is a special case of (1.3) with σ = · · · = σ r − = 1 and σ r = − 1. In particular, if all σ j = 1 ( j = 1 , , . . . , r ) in (1.3), then it becomesthe original (Kaneko–Tsumura) multiple T -value, which was introduced and studied by Kanekoand Tsumura in [16, 17]. 3 .2 Main Results The main results of this paper concern some weight sum formula of AMTVs and some dualityrelation for AMTVs. First, we extend the tools developed in our previous papers [26] and [27]to study the following weight sum formula of the AMTVs, W l ( k, r ) := X k k ··· + kr = k,k ,...,kr ≥ T ( k , . . . , k l − , k l , k l +1 , . . . , k r − , k r ) , (1.5)where 0 ≤ l ≤ r . Obviously, W ( k, r ) = W r ( k, r ) = W ( k, r ). Then we prove the followingtwo theorems concerning the duality of W l ( k, r ). Let N be the set of positive integers and N := N ∪ { } . Theorem 1.2.1. For any m, p ∈ N and l ∈ N , we have ( − m p X j =1 α p − j W l (2 j + 2 m + l − , m + l ) = ( − p m X j =1 α m − j W l (2 j + 2 p + l − , p + l ) , (1.6) where α n = ( − n π n / (2 n + 1)! . Theorem 1.2.2. For any m, p ∈ N and l ∈ N ,, we have ( − m p X j =1 α p − j W l (2 j +2 m + l − , m + l − 1) = ( − p m X j =1 α m − j W l (2 j +2 p + l − , p + l − . (1.7)Clearly, if l = 0, then the Theorems 1.2.1 and 1.2.2 above degenerate to [26, Theorem 1.1]and [26, Thm. 1.2], respectively.Next, for k = ( k , . . . , k r ) ∈ N r we often write k r = k , define its weight | k | := k + k + · · · + k r and adopt the following notation: −→ k j := ( k , k , . . . , k j ) , ←− k j := ( k r , k r − , . . . , k r +1 − j ) , |−→ k j | := k + k + · · · + k j , |←− k j | := k r + k r − + · · · + k r +1 − j with −→ k = ←− k := ∅ and |−→ k | = |←− k | := 0.Another purpose of present paper is to prove the following a more general duality relationfor general weighted sums involving AMTVs and binomial coefficients by using the method ofiterated integrals. Theorem 1.2.3. For positive integers k , . . . , k r and integers m, p ≥ , X j j ··· + j | k | +1= m +1+ | k | ,jℓ ≥ ∀ ℓ (cid:18) j | k | +1 + p − p (cid:19) × T (cid:18) r Cat l =1 n { j |←− k l − | +1 } ⋄ { j |←− k l − | +2 , , . . . , j |←− k l |− } ⋄ { j |←− k l | } o , j | k | +1 + p (cid:19) + ( − | k | +1 X j j ··· + j | k | +1= p +1+ | k | ,jℓ ≥ ∀ ℓ (cid:18) j | k | +1 + m − m (cid:19) T (cid:18) r Cat l =1 n { j |−→ k l − | +1 } ⋄ { j |−→ k l − | +2 , . . . , j |−→ k l |− } ⋄ { j |−→ k l | } o , j | k | +1 + m (cid:19) = r X l =1 ( − |←− k r − l | k l X j =1 ( − j − T (cid:16) { } m − , , ←− k r − l , j (cid:17) T (cid:16) { } p − , , −→ k l − , k l − j + 1 (cid:17) , (1.8) where ( { } − , 1) := ∅ , { l } m denotes the sequence obtained by repeating l exactly m times, ( a ⋄ { a , . . . , a r − } ⋄ a r ) := (cid:26) (¯ a , a , . . . , a r − , ¯ a r ) , r ≥ , ( a ) , r = 1 , and Cat rl =1 { s ( l ) } denotes the concatenated sequence ( s (1) , s (2) , . . . , s ( r )) . If r = 0 , then Cat l =1 { s ( l ) } := ∅ . Setting k = k = · · · = k r = 1 in Theorem 1.2.3, we obtain [26, Theorem 1.3]. In particular,setting k = k ∈ N and m = 0 in Theorem 1.2.3 we easily deduce the following corollary. Corollary 1.2.4. For positive integer k ≥ and integer p ≥ , X j j ··· + jk +1= p + k +1 ,jℓ ≥ ∀ ℓ T ( j , j , . . . , j k − , j k , j k +1 )= ( − k T (¯1 , { } k − , ¯1 , p + 1) + ( − k k X j =1 ( − j T (¯ j ) T ( { } p − , ¯1 , k − j + 1) , (1.9) where ( { } − , ¯1) := ∅ . We will prove Theorems 1.2.1-1.2.3 in the Sections 3 and 4. At the end of section 4, wepresent Conjecture 4.2.3, a parity conjecture on the MMVs which cannot be deduced from thewell-known parity principle for multiple polylogarithms proved by Panzer [20].In the last section, we investigate the structure of the Q -vector space AMTV w generated byall AMTVs of weight w . Conjecture 1.2.5. (=Conjecture 5.0.2) Let d w = dim Q AMTV w for all w ≥ . Set d = 1 .Then ∞ X w =0 d w t w = 11 − t − t − t . Namely, the dimensions form the tribonacci sequence { d w } w ≥ = { , , , , , , . . . } . Using Deligne’s theorem [6, Theorem 6.1] on colored MVZs of level four (i.e., special values ofmultiple polylogarithms at fourth roots of unity) we are able to verify Conjecture 1.2.5 rigorouslyfor w ≤ w = 6 by expressingevery AMTV in terms of Deligne’s basis (5.1) using the integer relation detecting algorithmPSLQ [2]. Here, in order to express every AMTV in terms of Deligne’s basis (5.1) we need tokeep 1500 digits of precision in MAPLE. Furthermore, in every weight w ≤ T ( k , . . . , k r ; σ , . . . , σ r ) with a suitable choice of thealternating signs ( σ , . . . , σ r ) ∈ {± } r for each choice of ( k , . . . , k r ) ∈ { , , } r . It would bevery interesting to determine the exact pattern of this basis in general.5 Iterated Integral Expressions In this section, we define a variant of Kaneko–Tsumura A-function with r -variables, and findsome iterated integral expressions of it and AMTVs. r -Variables Kaneko and Tsumura [16, 17] introduced the following a new kind of multiple polylogarithmfunctions of level twoA( k , . . . , k r ; x ) : = 2 r X 1, then T ( k , . . . , k r − , k r ) = i r A( k , . . . , k r ; i) . Similarly, we define a variant of (2.1) with r -variables byA( k , . . . , k r ; x , x , . . . , x r ):= 2 r X 2) + T (¯1 , ¯2) + T (¯2 , ¯1) . Similar to the duality relation of MZVs and MTVs, we may use the iterated integral ex-pression of AMTVs to derive their duality relation. Theorem 2.2.1. For any admissible k = ( k , . . . , k r ; σ , . . . , σ r ) , we define its dual by k ∗ := q (cid:0) ω k r − u σ r · · · ω k − u σ σ ··· σ r ω k − u σ σ σ ··· σ r (cid:1) (2.8) where u − := ω − , u := ω , and q is defined by (2.7) . Then T ( k ) = T ( k ∗ ) . (2.9) Proof. Applying the substitution t → − t t in (2.5), we get Z ω σ ω k − ω σ ω k − · · · ω σ r ω k r − = Z ω k r − u σ r · · · ω k − u σ ω k − u σ (2.10)7here u − := ω − and u := ω . By (2.4) we see that T ( k , . . . , k r ; σ , . . . , σ r ) = r Y ℓ =1 σ ′ ℓ ! Z ω σ σ σ ··· σ r ω k − ω σ σ ··· σ r ω k − · · · ω σ r ω k r − = r Y ℓ =1 σ ′ ℓ ! Z ω k r − u σ r ω k r − − u σ r − σ r · · · ω k − u σ σ σ ··· σ r by (2.10). So by (2.5) and (2.7) we have T ( k ) = r Y ℓ =1 σ ′ ℓ ! r Y ℓ =1 σ ′ ℓ =1 σ ′ ℓ T ( k ∗ ) = r Y ℓ =1 σ ′ ℓ ! T ( k ∗ ) = T ( k ∗ )by the definition (2.8). This completes the proof of the theorem.For example, T (¯1 , ¯3 , , ¯5) = − Z ω − ω ω ω − ω ω − ω = − Z ω u − ω u − ω u u − = − Z ω ω − ω ω − ω ω ω − = T ( { } , { ¯1 } , , ¯2 , ¯1) . Now, we establish some iterated integral identities of weighted sums involving AMTVs.Applying the changes of variables t j → − t r +1 − j t r +1 − j to (2.3) givesA( k , . . . , k r − , k r ; x , . . . , x r )= 4 r r Y j =1 ( − k j − Γ( k j ) x j Z D r log k − (cid:18) (1 − t r )(1 + t r − )(1 + t r )(1 − t r − ) (cid:19) (1 + t r ) − x (1 − t r ) · · · log k r − − (cid:18) (1 − t )(1 + t )(1 + t )(1 − t ) (cid:19) (1 + t ) − x r − (1 − t ) × log k r − (cid:18) − t t (cid:19) (1 + t ) − x r (1 − t ) dt dt · · · dt r . (2.11)Then replacing k r by k r + m − X k ··· + kr = k + r − ,k ,...,kr ≥ (cid:18) k r + m − m − (cid:19) A( k , . . . , k r − , k r + m − x , . . . , x r )= ( − k + m r x · · · x r ( m − k − Z D r log m − (cid:18) − t t (cid:19) log k − (cid:18) − t r t r (cid:19) dt · · · dt r [(1 + t ) − x r (1 − t ) ] · · · [(1 + t r ) − x (1 − t r ) ] . (2.12)8n the other hand, according to definition of AMTVs and using (2.2) and (2.3), by directcalculation we haveA( j , j , . . . , j r + l ; { } l , { i } r ) = 1i r T ( j , . . . , j l − , j l , j l +1 , . . . , j l + r − , j l + r ) (2.13)and A( j , j , . . . , j | k | +1 ; i , { } k r − , i , . . . , { } k − , i , { } k − , i)= 1i r +1 T (cid:18) r Cat l =1 n { j |←− k l − | +1 } ⋄ { j |←− k l − | +2 , . . . , j |←− k l |− } ⋄ { j |←− k l | } o , j | k | +1 (cid:19) . (2.14)Hence, from (2.12)-(2.14) we deduce the following two identities: W l ( k + r + l − , r + l )= X j j ··· + jr + l = k + r + l − ,j ,j ,...,jr + p ≥ T ( j , . . . , j l − , j l , j l +1 , . . . , j l + r − , j l + r )= ( − k + r − ( k − r Z D r + l log k − (cid:18) − t r + l t r + l (cid:19) (1 + t ) · · · (1 + t r ) t r +1 · · · t r + p dt · · · dt r + p (2.15)and p ! m !( − p + m + r +1 r +1 X j j ··· + j | k | +1= p +1+ | k | ,jℓ ≥ ∀ ℓ (cid:18) j | k | +1 + m − m (cid:19) × T (cid:18) r Cat l =1 n { j |←− k l − | +1 } ⋄ { j |←− k l − | +2 , , . . . , j |←− k l |− } ⋄ { j |←− k l | } o , j | k | +1 + m (cid:19) = Z log m (cid:18) − t t (cid:19) dt t (cid:18) dtt (cid:19) k − dt t (cid:18) dtt (cid:19) k − · · · dt t (cid:18) dtt (cid:19) k r − log p (cid:18) − t t (cid:19) dt t . (2.16) In this section, we give the proofs of Theorems 1.2.1 and 1.2.2, and establish some explicitevaluations for some special AMTVs and weighted sums involving AMTVs. Furthermore, wedefine a alternating convoluted T -values T ( k ¯ ⊛ l ) and a alternating version of Kaneko–Tsumura ψ -function (called Kaneko–Tsumura ¯ ψ -function), and study some explicit relations among the al-ternating convoluted T -values, Kaneko–Tsumura ¯ ψ -values and weighted sums involving AMTVs. T -Harmonic Sums and Multiple S -Harmonic Sums For k := ( k , . . . , k r ) ∈ N r , we put k r := ( k , . . . , k r ) . For positive integers m and n such that n ≥ m , we define D n,m := n ( n , n , . . . , n m ) ∈ N m | < n ≤ n < n ≤ · · · ≤ n m − < n m ≤ n o , if 2 ∤ m ; n ( n , n , . . . , n m ) ∈ N m | < n ≤ n < n ≤ · · · < n m − ≤ n m < n o , if 2 | m ,9 n,m := n ( n , n , . . . , n m ) ∈ N m | ≤ n < n ≤ n < · · · < n m − ≤ n m < n o , if 2 ∤ m ; n ( n , n , . . . , n m ) ∈ N m | ≤ n < n ≤ n < · · · ≤ n m − < n m ≤ n o , if 2 | m . Definition 3.1. ( [27, Defn. 1.1]) For positive integer m , define T n ( k m − ) := X n ∈ D n, m − m − ( Q m − j =1 (2 n j − − k j − (2 n j ) k j )(2 n m − − k m − , (3.1) T n ( k m ) := X n ∈ D n, m m Q mj =1 (2 n j − − k j − (2 n j ) k j , (3.2) S n ( k m − ) := X n ∈ E n, m − m − ( Q m − j =1 (2 n j − ) k j − (2 n j − k j )(2 n m − ) k m − , (3.3) S n ( k m ) := X n ∈ E n, m m Q mj =1 (2 n j − ) k j − (2 n j − k j , (3.4)where T n ( k m − ) := 0 if n < m , and T n ( k m ) = S n ( k m − ) = S n ( k m ) := 0 if n ≤ m . Moreover,for convenience sake, we set T n ( ∅ ) = S n ( ∅ ) := 1. We call (3.1) and (3.2) are multiple T -harmonicsums, and call (3.3) and (3.4) are multiple S -harmonic sums.Similar to the definition of convoluted T -values T ( k r ⊛ l s ) (see [27, Defn. 1.2]), we usethe MTHSs and MSHSs to define the alternating convoluted T -values T ( k r ¯ ⊛ l s ), which can beregarded as a alternating T -variant of Kaneko–Yamamoto MZVs ζ ( k ⊛ l ⋆ ) (for detail, see [18]),where k ≡ k r = ( k , . . . , k r ) and l ≡ l s = ( l , . . . , l s ). Definition 3.2. For positive integers m and p , the alternating convoluted T -values T ( k m ¯ ⊛ l p ) = 2 ∞ X n =1 T n ( k m − ) T n ( l p − )(2 n ) k m + l p ( − n , (3.5) T ( k m − ¯ ⊛ l p − ) = 2 ∞ X n =1 T n ( k m − ) T n ( l p − )(2 n − k m − + l p − ( − n , (3.6) T ( k m ¯ ⊛ l p − ) = 2 ∞ X n =1 T n ( k m − ) S n ( l p − )(2 n ) k m + l p − ( − n , (3.7) T ( k m − ¯ ⊛ l p ) = 2 ∞ X n =1 T n ( k m − ) S n ( l p − )(2 n − k m − + l p ( − n . (3.8)Here we allow k m + l p = 1 in (3.5), k m − + l p − = 1 in (3.6), k m + l p − = 1 in (3.7) and k m − + l p = 1 in (3.8).In order to better describe the main results, we need the following two lemmas. Lemma 3.1.1. (cf. [27, Theorem 2.1]) For positive integers m and n , the following identitieshold Z t n − log m (cid:18) − t t (cid:19) dt = 2(2 m )!2 n − m X j =0 ¯ ζ (2 j ) T n ( { } m − j ) , (3.9)10 t n − log m − (cid:18) − t t (cid:19) dt = − (2 m − n − m X j =1 ζ (2 j − T n ( { } m − j ) + S n ( { } m − ) , (3.10) Z t n − log m (cid:18) − t t (cid:19) dt = (2 m )!2 n m X j =1 ¯ ζ (2 j − T n ( { } m − j +1 ) + S n ( { } m ) , (3.11) Z t n − log m − (cid:18) − t t (cid:19) dt = − (2 m − n m − X j =0 ¯ ζ (2 j ) T n ( { } m − j − ) , (3.12) where ¯ ζ ( k ) stands for alternating Riemann zeta value denoted by ¯ ζ ( k ) := ∞ X n =1 ( − n − n k and ¯ ζ (0) := 12 ( k ∈ N ) . (3.13) Lemma 3.1.2. (cf. [27, Lemma 5.1]) Let A p,q , B p , C p ( p, q ∈ N ) be any complex sequences. If p X j =1 A j,p B j = C p and A p,p := 1 (3.14) hold, then B p = p X j =1 C j p − j X k =1 ( − k X i
Theorem 3.2.1. For any m, p ∈ N and l ∈ N , W l (2 p + 2 m + l − , m + l ) = 2( − m p − X j =0 ¯ ζ (2 p − − j ) T (cid:16)(cid:0) { } m − , l (cid:1) ¯ ⊛ (cid:0) { } j +2 (cid:1)(cid:17) , (3.16) W l (2 p + 2 m + l − , m + l ) = 2( − m p − X j =0 ζ (2 p − − j ) T (cid:16)(cid:0) { } m − , l (cid:1) ¯ ⊛ (cid:0) { } j +2 (cid:1)(cid:17) + ( − m T (cid:16)(cid:0) { } m − , l (cid:1) ¯ ⊛ (cid:0) { } p − (cid:1)(cid:17) , (3.17) W l (2 p + 2 m + l − , m + l − 1) = 2( − m − p − X j =0 ζ (2 p − − j ) T (cid:16)(cid:0) { } m − , l (cid:1) ¯ ⊛ (cid:0) { } j +1 (cid:1)(cid:17) + ( − m − T (cid:16)(cid:0) { } m − , l (cid:1) ¯ ⊛ { } p (cid:17) , (3.18)11 l (2 p + 2 m + l − , m + l − 1) = 2( − m − p − X j =0 ζ (2 p − − j ) T (cid:16)(cid:0) { } m − , l (cid:1) ¯ ⊛ (cid:0) { } j +1 (cid:1)(cid:17) . (3.19) Proof. Expanding the (1 + t j ) − into geometric series, then the formula (2.15) can be rewrittenin the form W l ( k + r + l − , r + l )= ( − k − r ( k − X 1, then W l (2 p + 2 m + l − , m + l )= 2( − m (2 p − ∞ X n =1 ( − n T n ( { } m − )(2 n ) l Z t n − log p − (cid:18) − t t (cid:19) dt, if r = 2 m − k = 2 p , then W l (2 p + 2 m + l − , m + l − − m (2 p − ∞ X n =1 ( − n T n ( { } m − )(2 n − l Z t n − log p − (cid:18) − t t (cid:19) dt, if r = 2 m − k = 2 p − 1, then W l (2 p + 2 m + l − , m + l − − m − (2 p − ∞ X n =1 ( − n T n ( { } m − )(2 n − l Z t n − log p − (cid:18) − t t (cid:19) dt. Then, with the help of integrals (3.9)-(3.12), we may easily deduce these desired formulas.Therefore, from (3.16), (3.19) and Lemma 3.1.2, we can get the following two theorems. Theorem 3.2.2. For any m, p ∈ N and l ∈ N , T (cid:16)(cid:0) { } m − , l (cid:1) ¯ ⊛ (cid:0) { } p (cid:1)(cid:17) = 2 ∞ X n =1 T n ( { } m − ) T n ( { } p − )(2 n ) l +1 ( − n = ( − m p X j =1 α p − j W l (2 j + 2 m + l − , m + l ) (3.20) where α n = ( − n π n / (2 n + 1)! for all n ∈ N . roof. Setting A j,p = 2¯ ζ (2 p − j ) , B j = ∞ X n =1 T n ( { } m − ) T n ( { } j − )(2 n ) l +1 ( − n and C p = ( − m W l (2 p + 2 m + l − , m + l )in Lemma 3.1.2 and using (3.16), we see that it suffices to prove Z ( j, p ) = α p − j where Z ( j, p ) := p − j X k =1 ( − k X i
Theorem 3.2.3. For any m, p ∈ N and l ∈ N , T (cid:16)(cid:0) { } m − , l (cid:1) ¯ ⊛ (cid:0) { } p − (cid:1)(cid:17) = 2 ∞ X n =1 T n ( { } m − ) T n ( { } p − )(2 n − l +1 ( − n = ( − m − p X j =1 α p − j W l (2 j + 2 m + l − , m + l − 1) (3.21) where α n = ( − n π n / (2 n + 1)! for all n ∈ N .Proof. Setting A j,p := 2¯ ζ (2 p − j ) , B j := 2 ∞ X n =1 T n ( { } m − ) T n ( { } j − )(2 n − l +1 ( − n and C p := ( − m − W l (2 p + 2 m + l − , m + l − m and p , the sums2 ∞ X n =1 T n ( { } m − ) S n ( { } p − )(2 n ) l +1 ( − n and 2 ∞ X n =1 T n ( { } m − ) S n ( { } p − )(2 n − l +1 ( − n can also be evaluated in terms of products of weighted sums W l ( k, r ) and α j − p . By elementarycalculations, we can get the following theorems.13 heorem 3.2.4. For any m, p ∈ N and l ∈ N , T (cid:16)(cid:0) { } m − , l (cid:1) ¯ ⊛ (cid:0) { } p − (cid:1)(cid:17) = 2 ∞ X n =1 T n ( { } m − ) S n ( { } p − )(2 n ) l +1 ( − n = ( − m W l (2 p + 2 m + l − , m + l ) − − m X ≤ i ≤ j ≤ p − α j − i ¯ ζ (2 p − − j ) W l (2 i + 2 m + l − , m + l ) . Theorem 3.2.5. For any m, p ∈ N and l ∈ N , T (cid:16)(cid:0) { } m − , l (cid:1) ¯ ⊛ (cid:0) { } p (cid:1)(cid:17) = 2 ∞ X n =1 T n ( { } m − ) S n ( { } p − )(2 n − l +1 ( − n = ( − m − W l (2 p + 2 m + l − , m + l − − − m − X ≤ i ≤ j ≤ p α j − i ¯ ζ (2 p + 1 − j ) W l (2 i + 2 m + l − , m + l − . Proofs of Theorems 1.2.1 and 1.2.2 . Changing ( m, p ) to ( p, m ) in (3.20) and (3.21), andusing the duality of series on the left hand sides, we may easily deduce the evaluations (1.6) and(1.7). Thus, we complete the proofs of Theorems 1.2.1 and 1.2.2. (cid:3) In this subsection, we introduce the multiple integrals associated with 3-labeled posets, anddefine the Kaneko–Tsumura ¯ ψ -function. We express AMTVs and Kaneko–Tsumura ¯ ψ -valuesin terms of multiple integral associated with 3-labeled posets, which implies that the Kaneko–Tsumura ¯ ψ -values can be expressed in terms of linear combination of MTVs. Further, weestablish some explicit relations between alternating convoluted T -values and Kaneko–Tsumura¯ ψ -values. The key properties of these integrals was first studied by Yamamoto in [28]. Definition 3.3. A 3 -poset is a pair ( X, δ X ), where X = ( X, ≤ ) is a finite partially ordered setand δ X is a map from X to {− , , } . We often omit δ X and simply say “a 3-poset X ”. The δ X is called the label map of X .Similar to 2-poset, a 3-poset ( X, δ X ) is called admissible if δ X ( x ) = 1 for all maximalelements and δ X ( x ) = 0 for all minimal elements x ∈ X . Definition 3.4. For an admissible 3-poset X , we define the associated integral I ( X ) = Z ∆ X Y x ∈ X ω δ X ( x ) ( t x ) , (3.22)where ∆ X = (cid:8) ( t x ) x ∈ [0 , X (cid:12)(cid:12) t x < t y if x < y (cid:9) and ω − ( t ) := 2 dt t , ω ( t ) := dtt and ω ( t ) := 2 dt − t . For the empty 3-poset, denoted ∅ , we put I ( ∅ ) := 1.14 roposition 3.3.1. For non-comparable elements a and b of a -poset X , X ba denotes the -poset that is obtained from X by adjoining the relation a < b . If X is an admissible -poset,then the -poset X ba and X ab are admissible and I ( X ) = I ( X ba ) + I ( X ab ) . (3.23)Note that the admissibility of a 3-poset corresponds to the convergence of the associatedintegral. We use Hasse diagrams to indicate 3-posets, with vertices ◦ and “ • σ ” correspondingto δ ( x ) = 0 and δ ( x ) = σ ( σ ∈ {± } ), respectively. For convenience, if σ = 1, replace “ • 1” by • and if σ = − 1, replace “ • − 1” by “ • ¯1”. For example, the diagram • ◦ ⑧⑧⑧⑧ • ❄❄❄❄ ¯1 ◦ ⑧⑧⑧⑧ ◦ ⑧⑧⑧⑧ • ❄❄❄❄❄❄❄ ¯1 ◦ ⑧⑧⑧⑧ ◦ ⑧⑧⑧⑧ represents the 3-poset X = { x , x , x , x , x , x , x , x } with order x < x > x < x 0, we define the Kaneko–Tsumura¯ ψ -function by ¯ ψ ( k , k . . . , k r ; s ) := 1Γ( s ) ∞ Z t s − sinh( t ) B( k , . . . , k r ; tanh( t/ dt, (3.27)where for x ∈ [ − , k , . . . , k r ; x ) := (cid:26) ( − m iA( k , . . . , k m − ; i x ) , if r = 2 m − − m A( k , . . . , k m ; i x ) , if r = 2 m. By an elementary calculation, we obtain the iterated integralB( k , . . . , k r ; x ) = Z x dt t (cid:18) dtt (cid:19) k − · · · dt t (cid:18) dtt (cid:19) k r − . According to definition and using the fact that if x = tanh( t/ 2) and s = p + 1 ∈ N then dx/x = dt/ sinh( t ) and 2 dx/ (1 − x ) = dt we deduce that¯ ψ ( k , k . . . , k r ; p + 1) = ( − p p ! Z log p (cid:18) − x x (cid:19) B( k , . . . , k r ; x ) x dx. (3.28)As an application, we can get the following theorem immediately. Theorem 3.3.2. For k = ( k , . . . , k r ) ∈ N r and integer p ≥ , we have ¯ ψ ( k ; p + 1) = 1 p ! I •◦◦ ◦ ♦♦♦♦♦♦♦ • ✖✖✖✖✖✖✖✖✖ • ✲✲✲✲✲✲✲✲✲ ( k , ¯1 ) p = I •◦◦ ◦ ⑧⑧⑧⑧⑧ ••• ❄❄❄❄❄ ( k , ¯1 ) p since there are exactly p ! ways to impose a total order on the p black vertices. Here ¯1 := ( { ¯1 } r ) . By Proposition 3.3.1 this implies the result that ¯ ψ ( k , . . . , k r ; p + 1) can be expressed as afinite sum of AMTVs. For example,¯ ψ (1 , 2; 2) = T (1 , ¯2 , 2) + T (1 , ¯1 , 3) + T (¯1 , ¯1 , ¯3) + T (¯1 , , ¯3) . Theorem 3.3.3. For positive integers m and p , ¯ ψ ( k m − ; 2 p ) = ( − m p − X j =0 ¯ ζ (2 p − − j ) T ( k m − ¯ ⊛ { } j +1 ) + ( − m T ( k m − ¯ ⊛ { } p ) , ¯ ψ ( k m − ; 2 p + 1) = ( − m p X j =0 ¯ ζ (2 p − j ) T ( k m − ¯ ⊛ { } j +1 ) , ¯ ψ ( k m ; 2 p ) = ( − m p − X j =0 ¯ ζ (2 p − − j ) T ( k m ¯ ⊛ { } j +2 ) , ¯ ψ ( k m ; 2 p + 1) = ( − m p − X j =0 ¯ ζ (2 p − − j ) T ( k m ¯ ⊛ { } j +2 ) + ( − m T ( k m ¯ ⊛ { } p +1 ) . roof. This follows immediately from the identity (3.28) and Lemma 3.1.1. We leave the detailto the interested reader.Comparing Theorem 3.2.1 and Theorem 3.3.3 with k = · · · = k r − = 1 and k r = l , weobtain the following theorem. Theorem 3.3.4. For any l, r ∈ N and p ∈ N , ¯ ψ ( { } r − , l ; p + 1) = ( − r W l ( p + r + l, r + l ) . Similar to [27, Theorem 5.3 and 5.4], using Lemma 3.1.2 and Theorem 3.3.3, we have thefollowing theorems. Theorem 3.3.5. For any k m − ∈ N m − and p ∈ N , T ( k m − ¯ ⊛ { } p +1 ) = ( − m p X j =1 α p − j (cid:16) ¯ ψ ( k m − ; 2 j + 1) + 2 ζ (2 j ) T ( k m − , k m − + 1) (cid:17) . Theorem 3.3.6. For any k m ∈ N m and p ∈ N , T ( k m ¯ ⊛ { } p ) = ( − m p X j =1 α p − j ¯ ψ ( k m ; 2 j ) . The proofs of Theorems 3.3.5 and 3.3.6 are completely similar to the proofs of [27, The-orem 5.3 and 5.4]) and are thus omitted. Using Theorems 3.3.3, 3.3.5 and 3.3.6, we can alsoevaluate explicitly the alternating convoluted T -values T ( k m − ¯ ⊛ { } p ) and T ( k m ¯ ⊛ { } p +1 )in terms of Kaneko–Tsumura ¯ ψ -values and (alternating) Riemann zeta values. This implies thefollowing result. Corollary 3.3.7. For k = ( k , . . . , k r ) ∈ N r and positive integer p , the alternating convoluted T -value T ( k ¯ ⊛ { } p ) can be expressed as a Z -linear combination of products of alternating MTVs and alternating Riemann zeta values. Now, we end this section by the following theorem. Theorem 3.3.8. For any positive integers l , l and k m ∈ N m , T ( k m ¯ ⊛ ( l , l )) = f ( k m , l , l ) + ( − [( m +1) / I • ¯1 ⑧⑧⑧ ◦ ◦ • ¯1 ⑧⑧⑧ ◦ ◦ ⑧⑧⑧ ◦ ⑧⑧⑧ ◦ ◦ ✴✴✴✴ • ⑧⑧⑧ ◦ ◦ k k m l l where f ( k m , l , l ) = 0 if m is even and f ( k m , l , l ) = 2( − [( m +1) / ¯ ζ ( l ) T ( k , . . . , k m − , k m + l ) if m is odd.Proof. The proof of Theorem 3.3.8 is completely similar to the proof of [27, Theorem 4.5] andis thus omitted. We leave the detail to the interested reader.17 .4 Some Special Values of AMTVs and Weighted Sums Next, we consider some specific cases. Setting l = m = 1 in (1.7) yields T (¯1 , { } p − , ¯1) = ( − p − p X j =1 α p − j W (2 j, . (3.29)According to definition, we see that W (2 j, 2) = X k k j,k ,k ≥ T (¯ k , ¯ k )and T (¯ k , ¯ k ) = 4 X 1, and t ( k ) is the alternating t -value defined by t ( p ) := ∞ X n =1 ( − n − ( n − / p ( p ∈ N ) . In particular, t (2) = 4 G where G := ∞ X n =1 ( − n − / (2 n − is Catalan’s constant, and t (2 k + 1)is related to the Euler number E k by t (2 k + 1) = ( − k E k π k +1 k )! ( k ≥ , where sec( x ) = ∞ X k =0 ( − k E k (2 k )! x k . In particular, we have E = 1 , E = − , E = 5 , E = − 61 and E = 1385.18learly, T (¯ k ) = − k − t ( k ) ( k ∈ N ) . Therefore, applying (3.30) we obtain W (2 j, 2) = − j − j X k =1 t (2 k )¯ ζ (2 j − k ) . (3.31)Taking j = 1 , W (2 , 2) = − t (2) , W (4 , 2) = − t (4) − t (2) ζ (2)and W (6 , 2) = − t (2) ζ (4) − t (4) ζ (2) − t (6) . Further, plugging (3.31) into (3.29), we can get the following theorem. Theorem 3.4.1. For any p ∈ N , we have T (¯1 , { } p − , ¯1) = ( − p X ≤ k ≤ j ≤ p t (2 k ) ζ (2 j − k ) α j − p j − = T ( { } p − , ¯2) . (3.32) Proof. The first equality follows immediately from (3.29) and (3.31). The second equality followsfrom the duality relation T ( { } p − , , { } r − , 1) = T ( { } r − , p + 1) (see [26, (4.6)]).Setting p = 2 and 3 we obtain T (¯1 , , , ¯1) = 18 t (4) − t (2) ζ (2) = T (1 , , ¯2) T (¯1 , , , , , ¯1) = − t (2) ζ (4) + 332 t (4) ζ (2) − t (6) = T (1 , , , , ¯2) . Moreover, in [26, Theorem 2.3 and (4.10)], the first author shows that T ( { } r − , ¯ k ) = r X j =1 ( − j − T ( { } r − j − , ¯1) W ( k + j − , j ) , (3.33) W ( k + r − , r ) = r X j =1 ( − j − T ( { } r − j − , ¯1) T ( { } j − , ¯ k ) , (3.34)where k, r are positive integers and ( { } − , 1) := ∅ . If we set r = 2 p − k = 2 in (3.33),and r = 2 p and k = 2 in (3.34), then we get T ( { } p − , ¯2) = p − X j =1 ( − j − T ( { } p − − j , ¯1) W ( j + 1 , j )= p X j =1 T ( { } p − j − , ¯1) W (2 j, j − − p − X j =1 T ( { } p − j − , ¯1) W (2 j + 1 , j ) (3.35)19nd W (2 p + 1 , p ) = p X j =1 ( − j − T ( { } p − j − , ¯1) T ( { } j − , ¯2)= p X j =1 T ( { } p − j , ¯1) T ( { } j − , ¯2) − p X j =1 T ( { } p − j − , ¯1) T ( { } j − , ¯2) . (3.36)Furthermore, the first author proves in [26, Corollary 2.5] that for any p ∈ N , the weightedsums W (2 p + 1 , p ) can be expressed explicitly in terms of (alternating) Riemann zeta values byproviding an explicit formula. Hence, applying (3.32), (3.35) and the identity (see [26, (2.15)]) T ( { } r − , ¯1) = ( − r r ! (cid:16) π (cid:17) r , we arrive at the following result. Theorem 3.4.2. For any p ∈ N , the two AMTVs T (¯1 , { } p − , ¯1) and T ( { } p − , , and theweighted sums W (2 p, p − can be expressed as a Q -linear combination of products of thealternating t -values and the Riemann zeta values. For example, we compute the following cases T (1 , ¯2) = − ζ (3) + π t (2) = T (¯1 , , ¯1) ,T (1 , , , ¯2) = 3116 ζ (5) − πt (4) − π t (2) = T (¯1 , , , , ¯1) , and W (4 , 3) = 18 t (4) + t (2) ζ (2) − πζ (3) ,W (6 , 5) = − t (6) − π ζ (3) − t (2) ζ (4) + 3132 πζ (5) . Hence, from Theorems 3.4.1 and 3.4.2, we can conclude that for any positive integer p , T (¯1 , { } p − , ¯1) = T ( { } p − , ∈ Q [ t (1) , ζ (2) , t (2) , ζ (3) , t (3) , ζ (4) , . . . ] . Further, setting r = 2 p − k = 2 in (3.34) we find W (2 p, p − 1) = p − X j =1 ( − j − T ( { } p − j − , ¯1) T ( { } j − , ¯2) . (3.37)Very recently, the first author and Wang prove in [24, Theorem 4.9] that for any positiveintegers k , k , k and ( σ , σ , σ ) ∈ {± } with ( k , σ ) = (1 , 1) and ( k , σ ) = (1 , T -values(1 + σ σ ( − k + k + k ) T ( k , k , k ; σ , σ , σ )can be expressed in terms of combinations of (alternating) double M -values and single M -values,and give explicit though very complicated formula. Hence, from Theorem 1.2.2 we can obtainthe following theorem. 20 heorem 3.4.3. For positive integer p , the two AMTVs T (1 , ¯1 , { } p − , ¯1) and T ( { } p − , ¯3) (3.38) can be expressed as a Q -linear combinations of products of (alternating) double M -values andsingle M -values.Proof. Setting m = 1 and l = 2 in Theorem 1.2.2 yields W (2 p + 1 , p + 1) = T (1 , ¯1 , { } p − , ¯1) = ( − p − p X j =1 α p − j W (2 j + 1 , . From [24, Theorem 4.9] with σ = 1 , σ = σ = − 1, we know that the triple T -values T ( k , ¯ k , ¯ k )are reducible to combinations of double M -values and single M -values. Hence, using the defini-tion (1.5), we have W (2 k + 1 , 3) = X k k k k +1 ,k ,k ,k ≥ T ( k , ¯ k , ¯ k ) . Then, using the well-known duality relation (see [26, (4.6)]) T ( { } p − , , { } r − , 1) = T ( { } r − , p + 1)we obtain the desired description.For example, we calculate the following cases: W (3 , 3) = − π ,W (5 , 3) = − π − π 24 log (2) + π 24 log (2) + π Li (1 / 2) + π ζ (3) ,W (7 , 3) = − π − π 192 log (2) + π 192 log (2) + π (1 / 2) + 7 π 64 log(2) ζ (3) , + π T (1 , ¯5) + T (2 , ¯4) + T (3 , ¯3) + T (4 , ¯2) + T (5 , ¯1)) , and T (1 , ¯1 , ¯1) = T (¯3) = − π ,T (1 , ¯1 , , , ¯1) = T (1 , , ¯3) = 121 π π 24 log (2) − π 24 log (2) − π Li (1 / − π ζ (3) ,T (1 , ¯1 , { } , ¯1) = T ( { } , ¯3) = − π π 576 log (2) − π 576 log (2) − π 24 Li (1 / − π 192 log(2) ζ (3) + π T (1 , ¯5) + T (2 , ¯4) + T (3 , ¯3) + T (4 , ¯2) + T (5 , ¯1)) . In this section we prove the duality formula in Theorem 1.2.3 and find some duality relations ofAMTVs by using the method of iterated integrals.21 .1 Proof of A Duality Formula of Weighted Sums To prove Theorem 1.2.3, we need the following two lemmas. The first one follows quickly fromthe general theory of Chen’s iterated integrals. Lemma 4.1.1. (cf. [4, (1.6.1-2)]) If f i ( i = 1 , . . . , m ) are integrable real functions, the followingidentity holds: g ( f , . . . , f m ) + ( − m g ( f m , . . . , f ) = m − X i =1 ( − i − g ( f i , f i − , · · · , f ) g ( f i +1 , f i +2 · · · , f m ) , where g ( f , . . . , f m ) is defined by g ( f , . . . , f m ) := Z For any k = ( k , . . . , k r ) ∈ N r and integer p ≥ , Z log p (cid:18) − t t (cid:19) dt t (cid:18) dtt (cid:19) k − dt t (cid:18) dtt (cid:19) k − · · · dt t (cid:18) dtt (cid:19) k r − = ( − p + r p !2 r T ( { } p − , , k , . . . , k r − , k r ) , (4.1) where ( { } − , 1) := ∅ .Proof. First, we note the fact that A( { } p ; x ) = 1 p ! (A(1; x )) r = ( − p p ! log p (cid:18) − x x (cid:19) (see [16,Lemma 5.1 (ii)] or [26, (2.14)]). Then according to definition, we havelog p (cid:18) − x x (cid:19) = ( − p p !2 p X For any k = ( k , . . . , k r ) ∈ N r and integers p, m ≥ , Z log p (cid:18) − t t (cid:19) dt t (cid:18) dtt (cid:19) k − dt t (cid:18) dtt (cid:19) k − · · · dt t (cid:18) dtt (cid:19) k r − log m (cid:18) − t t (cid:19) dt t + ( − | k | + Z log m (cid:18) − t t (cid:19) dt t (cid:18) dtt (cid:19) k r − dt t (cid:18) dtt (cid:19) k r − − · · · dt t (cid:18) dtt (cid:19) k − log p (cid:18) − t t (cid:19) dt t = C m,p,r r X l =1 ( − |←− k r − l | k l X j =1 ( − j − T (cid:16) { } m − , , ←− k r − l , j (cid:17) T (cid:16) { } p − , , −→ k l − , k l − j + 1 (cid:17) , where C m,p,r := ( − m + p + r +1 m ! p !2 r +1 . roof. Let J p ( k , . . . , k r ) be the integral on the left hand of (4.1). Using Lemma 4.1.1, we deduce Z log p (cid:18) − t t (cid:19) dt t (cid:18) dtt (cid:19) k − dt t (cid:18) dtt (cid:19) k − · · · dt t (cid:18) dtt (cid:19) k r − log m (cid:18) − t t (cid:19) dt t = k r X l =1 ( − l − J m ( l ) J p ( k , . . . , k r − , k r − l + 1)+ ( − k r k r − X l =1 ( − l − J m ( k r , l ) J p ( k , . . . , k r − , k r − − l + 1)+ ( − k r + k r − k r − X l =1 ( − l − J m ( k r , k r − , l ) J p ( k , . . . , k r − , k r − − l + 1)+ · · · + ( − k r + k r − + ··· + k k X l =1 ( − l − J m ( k r , k r − , . . . , k , l ) J p ( k − l + 1)+ ( − | k | Z log m (cid:18) − t t (cid:19) dt t (cid:18) dtt (cid:19) k r − dt t (cid:18) dtt (cid:19) k r − − · · · dt t (cid:18) dtt (cid:19) k − log p (cid:18) − t t (cid:19) dt t . We can now complete the proof by applying the identity J p ( k , . . . , k r ) = ( − p + r p !2 r T ( { } p − , , k , . . . , k r − , k r )with a straightforward calculation, Proof of Theorem 1.2.3 . Plugging (2.16) into Theorem 4.1.3 we get the desired result (1.8). (cid:3) Further, using (2.12) and Lemma 4.1.1, we can also get the following theorem. Theorem 4.1.4. For positive integers m, k and r , X k ··· + kr = k + r − ,k ,...,kr ≥ (cid:18) k r + m − m − (cid:19) A( k , . . . , k r − , k r + m − x , . . . , x r − , x r )+ ( − r X k ··· + kr = m + r − ,k ,...,kr ≥ (cid:18) k r + k − k − (cid:19) A( k , . . . , k r − , k r + k − x r , . . . , x , x )= r − X j =1 ( − j − A( { } r − − j , m ; x j +1 , x j +2 , . . . , x r )A( { } j − , k ; x j , x j − , . . . , x ) . (4.3) Proof. This follows immediately from the (2.12) and Lemma 4.1.1. We leave the detail to theinterested reader. 23n particular, if r = 2 then we obtain the decompositionA( k ; x )A( m ; x ) = X k k k +1 ,k ,k ≥ (cid:18) k + m − m − (cid:19) A( k , k + m − x , x )+ X k k m +1 ,k ,k ≥ (cid:18) k + k − k − (cid:19) A( k , k + k − x , x ) . It is clear that we can find a lot of duality relations of (alternating) MTVs from Theorem4.1.4. For example, setting x = x = · · · = x r = 1 and m, k ≥ r − X j =1 ( − j − T ( { } r − − j , m ) T ( { } j − , k ) = X | k r | = k + r − (cid:18) k r + m − m − (cid:19) T ( k r − , k r + m − − r X | k r | = m + r − (cid:18) k r + k − k − (cid:19) T ( k r − , k r + k − , where k r = ( k , . . . , k r ) ∈ N r as before. Similarly, setting x = x = · · · = x r = − r − X j =1 ( − j − T ( { } r − − j , ¯ m ) T ( { } j − , ¯ k ) = X | k r | = k + r − (cid:18) k r + m − m − (cid:19) T ( k r − , k r + m − − r X | k r | = m + r − (cid:18) k r + k − k − (cid:19) T ( k r − , k r + k − . We can also find many interesting results about AMTVs from Theorem 1.2.3. For example,setting m = p = 0 in (1.8) yields the following corollary. Corollary 4.2.1. For any k = ( k , . . . , k r ) ∈ N r , we have T (cid:18) r Cat l =1 { ⋄ { } k r +1 − l − ⋄ } , ¯1 (cid:19) + ( − | k | +1 T (cid:18) r Cat l =1 { ⋄ { } k l − ⋄ } , ¯1 (cid:19) = r X l =1 ( − |←− k r − l | k l X j =1 ( − j − T (cid:16) ←− k r − l , j (cid:17) T (cid:16) −→ k l − , k l − j + 1 (cid:17) . (4.4)In particular, if putting r = 1 and k = 2 p + 1 ( p ≥ T (¯1 ⋄ { } p − ⋄ ¯1 , ¯1) = ( − p T ( p + 1) + p X j =1 ( − j − T (¯ j ) T (2 p + 2 − j ) . (4.5)Further, setting p = 0 and 1 we get T (1 , ¯1) = 34 ζ (2) , T (¯1 , , ¯1 , ¯1) = − t (2) + 4516 ζ (4) . We now turn to a duality type theorem of AMTVs.24 heorem 4.2.2. For any ( k , . . . , k r ) ∈ N r and integer p ≥ , T ( { } p − , , k , . . . , k r , 1) = T (1 ⋄ { } k r − ⋄ , . . . , ⋄ { } k − ⋄ , p + 1) , (4.6) where ( { } − , 1) := ∅ .Proof. Setting k r = 1 in (4.1) we get( − p + r p !2 r T ( { } p − , , k r − , 1) = Z log p (cid:18) − t t (cid:19) dt t (cid:18) dtt (cid:19) k − dt t · · · (cid:18) dtt (cid:19) k r − − dt t . Then changes of variables t j → − t r +1 − j t r +1 − j in the above identity yield( − p + r p !2 r T ( { } p − , , k r − , k + ··· + k r − − r +1 Z dt t (cid:18) dt − t (cid:19) k r − − · · · dt t (cid:18) dt − t (cid:19) k − log p ( t ) dt t = ( − p + r p !2 r T (1 ⋄ { } k r − − ⋄ , . . . , ⋄ { } k − ⋄ , p + 1)by (2.3). We can now complete the proof of the theorem by replacing r by r + 1.In particular, if k = · · · = k r = 1, then we obtain the well-know result T ( { } p − , , { } r , 1) = T ( { } r , p + 1) . If letting r = 1 , k = k and p = 0, then T (¯1 ⋄ { } k − ⋄ ¯1 , ¯1) = T ( k, ¯1) . (4.7)Hence, if putting k = 2 p + 1 ( p ≥ 0) and using (4.5) gives T (2 p + 1 , ¯1) = ( − p T ( p + 1) + p X j =1 ( − j − T (¯ j ) T (2 p + 2 − j ) . (4.8)On the other hand, from [23, Corollary 3.4], we have ∞ X n =1 h (2 p ) n n ( − n − = 2 p − T (2 p, ¯1) = p p +1 t (2 p + 1) − p X k =1 t (2 k − t (2 p − k + 2) . (4.9)Then applying (4.7) and the relation T (¯ k ) = − k − t ( k ) ( k ∈ N ), we obtain T (¯1 , { } p − , ¯1 , ¯1) = 2 pT (2 p + 1) − p X k =1 T (2 k − T (2 p − k + 1) ( p ∈ N ) . (4.10)As an example, setting p = 1 yields T (¯1 , ¯1 , ¯1) = 72 ζ (3) − π t (2) . orollary 4.2.3. For any k = ( k , . . . , k r ) ∈ N r , T ( k , ¯1) + ( − | k | T ( ←− k , ¯1) = r X l =1 ( − |←− k r − l | k l X j =1 ( − j − T (cid:16) ←− k r − l , j (cid:17) T (cid:16) −→ k l − , k l − j + 1 (cid:17) . (4.11) Proof. This follows immediately from the Corollary 4.2.1 and Theorem 4.2.2 with p = 0.We end this section by a parity conjecture for AMMVs based on our computations. Conjecture 4.2.4. For composition k = ( k , k , . . . , k r ) and ε = ( ε , . . . , ε r ) ∈ {± } r and σ = ( σ , . . . , σ r ) ∈ {± } r with ( k r , σ r ) = (1 , , − r Y j =1 (cid:16) ( − k j +1 sign( σ j + ε j + 1) (cid:17) M σ ( k ; ε ) can be expressed in terms of products of AMMVs of lower depth. For example, we have the following cases of double M -values M , − (ˇ2 , ˇ6) = − π ζ (3) − π ζ (5) − π ζ (7) + 2 G ,M − , (ˇ2 , ˇ6) = − π G + 5 π π ζ (5) + 381 π ζ (7) + 2 G ,M − , (ˇ2 , 6) = − π G − π G − π G + 381 π ζ (7) − G ,M − , (ˇ6 , 2) = 35 π ζ (3) + 31 π ζ (5) − π G + 381 π ζ (7) − G ,M − , − (ˇ2 , 6) = π G + π G + 4 π G − G ,M − , − (ˇ6 , 2) = π G + π G + 11 π G − G . Here G m = X n ≥ ( − n (2 n + 1) m is the generalized Catalan’s constant. Clearly, G m = t (2 m ) / m = − T (2 m ) / 2. In particular, G = G is Catalan’s constant. Remark 4.2.5. It should be emphasized that Panzer proves a general parity result on multiplepolylogarithms in [20, Theorem 1.3]. Let AMMV w be the Q -vector space generated by all theAMMVs of weight w and denote all its subspaces similarly. According to the definition of coloredMZVs of level four, we know that η j ∈ { e πi/ , e πi , e πi/ , e πi } = {± , ± i } . Hence, for coloredMZVs of level four, we have η n j j := (cid:26) {± , ± i ( − ( n j − / } if n j odd; { , ± ( − n j / } if n j even,and Li k ,k ,...,k r ( η , . . . , η r ) = X 1) + Li m,n ( − i, − (cid:17) . By Deligne’s result [6, Theorem 6.1] every CMVZ of depth 2 and weight 8 is a Q -linear combi-nations of double logarithm values Li m,n ( i, 1) and products of (2 πi ) p Li − p ( i ) (0 ≤ p ≤ Li m,n ( i, 1) with m + n = 8 can be reducedin the above sense (cf. [3]). Thus, our Conjecture 4.2.3 should not be a consequence of Panzer’sparity principle for multiple polylogarithms. We first relate AMTVs to colored MZVs via the following mechanism. For all k ∈ N r , η ∈ {± } r ,if ( k r , η r ) = (1 , 1) then T ( k ; η ) = 2 r · X 1. Thus the Q -vector space AMTV w generatedby all AMTVs of weight w lies in ( CMZV ,w ∪ i CMZV ,w ) ∩ R .We remark first that CMZV ,w ∩ R does not contain AMTV w in general. In fact, in weightone we have only one AMTV, i.e., T (¯1) = 2 ∞ X n =1 ( − n n − − π . On the other hand, CMZV , is generated by the three colored MZVs Li ( i ) = − 12 log 2 + π i, Li ( − i ) = − 12 log 2 − π i, Li ( − 1) = − log 2 . Therefore, CMZV , ∩ R = (log 2) Q . But it is well-known that log 2 and π are linearly independentover Q . Indeed, if log 2 = rπ for some nonzero r ∈ Q , then2 = e rπ = ( − ri ∈ Q . − ri is transcendental by Gelfond–Schneider theorem [10] since ri is algebraic butnot rational. This contradiction implies that AMTV CMZV , ∩ R . Set Li ∅ = 1. Deligne shows in [6, Theorem 6.1] that CMZV ,w can be generated by B w := n (2 πi ) p Li k ( i, , . . . , 1) : p + | k | = w o , (5.1)where p ∈ N and k runs through compositions of positive integers or the empty set. Forexample, setting L k = Li k ( i, { } d − ) for all k ∈ N d , we have B := (cid:26) πi, L = − 12 log 2 + π i (cid:27) or B ′ := n πi, log 2 o ; B := n (2 πi ) , (2 πi ) L , L , L , o or B ′ := n π , ( π log 2) i, G , log (2) o ; B := n (2 πi ) , (2 πi ) L , (2 πi ) L , (2 πi ) L , , L , L , , , L , , L , o or B ′ := n π i, π log 2 , πG , ( π log i, ζ (3) , log , L , , L , o . Similarly, denote by TB w the conjectural basis of Q -vector space generated by all AMTVs ofweight w . Theorem 5.0.1. For w ≤ , the follow set TB w and TB w ( Z ) generates the space AMTV w : TB = TB ( Z ) = (cid:8) T (¯1) (cid:9) , TB = TB ( Z ) = (cid:8) T (1 , ¯1) , T (¯2) (cid:9) , TB = TB ( Z ) = (cid:8) T (3) , T (1 , ¯2) , T (¯2 , ¯1) , T (1 , , ¯1) (cid:9) , TB = (cid:8) T (1 , ¯3) , T (2 , ¯2) , T (1 , , ¯2) , T (¯3 , ¯1) , T (1 , ¯2 , ¯1) , T (¯2 , , ¯1) , T (1 , , , ¯1) (cid:9) , TB ( Z ) = (cid:8) T (1 , , , ¯1) , T (¯2 , , T (¯1 , , T (1 , ¯3) , T (¯2 , , ¯1) , T (3 , ¯1) , T (¯4) (cid:9) , TB = ( T (2 , ¯3) , T (¯3 , ¯2) , T (1 , , ¯3) , T (1 , , ¯2) , T (¯2 , , ¯2) , T (1 , , , ¯2) , T (1 , ¯3 , ¯1) ,T (¯3 , , ¯1) , T (¯2 , ¯2 , ¯1) , T (1 , , ¯2 , ¯1) , T (1 , ¯2 , , ¯1) , T (¯2 , , , ¯1) , T (1 , , , , ¯1) ) , TB ( Z ) = ( T (¯5) , T (5) , T (¯4 , ¯1) , T (¯3 , ¯2) , T (¯3 , , T (1 , ¯4) , T (1 , , T (3 , ¯2) ,T (¯3 , ¯1 , ¯1) , T (¯2 , ¯2 , ¯1) , T (¯2 , ¯1 , , T (¯2 , , , T (¯1 , , ) . If a variation of Grothendieck’s period conjecture [6, Conj. 5.6] holds then TB w is a basis of AMTV w over Q . Furthermore, the bases TB w ( Z ) above are all integral basis, namely, everyAMTV of weight w is a Z -linear combination of the basis elements in TB w ( Z ) for all w ≤ .Proof. For an admissible composition k = ( k , . . . , k r ; σ , . . . , σ r ) of weight w we can set T ( k ) = σ σ . . . σ rr Z p ( k ) := σ σ . . . σ rr Z ω i . . . ω i w . by (2.6). Define α ( k ) := ♯ { j : i j = 0 } , β ( k ) := ♯ { j : i j = 1 } , γ ( k ) := ♯ { j : i j = − } . Then from the duality relation (2.10) we see that α ( k ) = β ( k ∗ ) , β ( k ) = α ( k ∗ ) , γ ( k ) = γ ( k ∗ ) . α ( k ) + α ( k ∗ ) = w − γ ( k ) = w − γ ( k ∗ ) . So to compute the vector space AMTV w we only need to consider AMTVs T ( k ) with2 α ( k ) ≤ w − γ ( k ) . Furthermore, if the equality holds in the above, then from the two dual values we only choosethe one so that the first ω in its iterated integral form appears earlier. For example, by (2.10) T (¯1 , ¯1) = − Z ω ω − = − Z ω − ω = T (¯2) , T (1 , 2) = T (3) ,T (1 , ¯1 , ¯1) = T (¯3) , T (¯1 , , ¯1) = T (1 , ¯2) , T ( { ¯1 } ) = T (2 , ¯1) , T (¯1 , 2) = T (¯2 , ¯1) . Clearly, there are exactly 4 · w − weight w AMTVs if we consider their iterated integral rep-resentations. Furthermore, the self-dual values are determined by the first half 1-forms intheir iterated integral representations (and the middle 1-form must be ω − if the weight w is odd). So there are 2 · [ w/ − such values. Therefore, by duality we only need to compute N w := 2 · w − + 3 [ w/ − values in AMTV w to determine its dimension. We have N := 3, N := 7, N := 21, N := 57.Now, straight-forward computation leads quickly to the conclusions in weight w ≤ T (¯1) = − π T (2) = 2 T (1 , ¯1) , T (¯2) = − G , T (1 , ¯1) = π T (¯3) = 3 T (1 , , ¯1) , T (¯1 , ¯2) = 6 T (1 , , ¯1) − T (¯2 , ¯1) , T (2 , ¯1) = T (3) − T (1 , ¯2) . By duality, T (¯1 , ¯1) and the other five weight 3 AMTVs can be found easily. We see immediatelythat the spans of TB and TB in the theorem generate the space AMTV and AMTV over Q ,respectively. Moreover, AMTV ⊗ Q ( i ) = (cid:10) ˜ π , L (cid:11) Q ( i ) and AMTV ⊗ Q ( i ) = (cid:10) ˜ π , ˜ πL , L , , L (cid:11) Q ( i ) , where ˜ π = 2 πi and L k = Li k ( i, , . . . , . So the bases TB and TB are linearly independentover Q if we assume [6, Conj. 5.6] which implies that Deligne’s basis (5.1) is linearly independentover Q ( i ) for all for w .In weight 4, using the computation from [30], we find by Maple that AMTV can be spannedover Q ( i ) by the seven elements from Deligne’s basis (5.1): n ˜ π , ˜ π L , ˜ πL , ˜ πL , , L , L , , L , o which is conjecturally linearly independent over Q ( i ). Further, it is straight-forward to changethis basis to the basis TB in the theorem. To save space, we set a = T (1 , , , ¯1) , a = T (¯2 , , a = T (¯1 , , a = T (1 , ¯3) ,a = T (¯2 , , ¯1) , a = T (3 , ¯1) , a = T (¯4) . Then T (4) = 8 a , T (¯2 , ¯2) = 9 a − a + 18 a , T (¯1 , ¯2 , ¯1) = 24 a − a − a ,T (2 , 2) = 4 a , T (¯1 , ¯3) = 3 a − a − a , T (1 , , ¯2) = 3 a − a + 6 a , (¯2 , ¯1 , ¯1) = 2 a , T (2 , ¯2) = 12 a − a − a , T (¯3 , ¯1) = 9 a − a − a ,T (1 , 3) = 6 a − a , T (¯1 , , ¯2) = 12 a − a , T (2 , , ¯1) = 2 a + 6 a − a ,T (1 , , ¯1) = − a − a + 6 a , T (¯1 , , ¯1) = − a + a + 2 a . All the other 15 weight 4 AMTVs can be obtained by duality Theorem 2.2.1, for e.g.: T (1 , ¯1 , 2) = a , T (¯1 , ¯1 , ¯2) = T (¯1 , , ¯1) , T (¯1 , , , ¯1) = T (1 , , ¯2) , T (¯1 , , ¯1 , ¯1) = T (3 , ¯1) . The last two equations are consistent with (3.32) and (4.7), too.For weight 5, similar computation can be carried out using Au’s Mathematica package [1,Appedix A] containing the explicit expressions of all values in CMZV , in terms of 32 basiselements. It can be further found with Maple that AMTV is contained in a co-dimension one Q ( i )-subspace of the space generated by the following 14 elements in the Deligne basis:˜ π , ˜ πL , , ˜ πL , , ˜ πL , ˜ π L , , ˜ π L , ˜ π L , L , L , , L , , L , , , L , , , L , , L , , . Finally, it can be verified that AMTV can be generated by the 13 elements in TB . To savespace, set b := T (1 , , , , ¯1) , b := T (2 , , , ¯1) , b := T (1 , , , ¯1) , b := T (¯1 , , , ¯1) ,b := T (¯2 , , , ¯1) , b := T (1 , , ¯1) , b := T (1 , ¯3 , ¯1) , b := T (3 , ¯1 , ¯1) ,b := T (¯3 , ¯1 , ¯1) , b := T (¯2 , ¯2 , ¯1) , b := T (¯1 , , b := T (1 , ¯4) , b := T (1 , . Then T (¯5) = 25 b , T (¯3 , 2) = 2 b + 4 b − b + 240 b + 21 b − b + 2 b ,T (5) = 2 b − b + 7 b , T (¯2 , 3) = − b − b + 8 b + 35 b − b − b − b ,T (2 , 3) = 2 b + 2 b − b , T (¯1 , ¯2 , ¯2) = − b − b + 12 b − b + 3 b + 4 b ,T (2 , ¯3) = − b + 4 b + b , T (2 , ¯1 , ¯2) = 180 b − b + 20 b − b + 2 b + 4 b + 2 b ,T (3 , 2) = 6 b , T (1 , ¯1 , 3) = 9 b + 3 b − b − b + 3 b ,T (¯1 , ¯4) = − b + 6 b − b + 2 b , T (¯1 , , ¯2) = 2 b + 3 b + 12 b − b − b ,T (¯3 , ¯2) = 2 b − b + 60 b − b , T (¯1 , , ¯1) = − b − b − b + 5 b + 2 b ,T (¯2 , ¯3) = 60 b − b + 5 b − b , T (¯2 , ¯1 , 2) = 4 b − b + 28 b − b + 2 b ,T (3 , ¯2) = 12 b − b − b , T (2 , ¯2 , ¯1) = − b − b + 4 b + 28 b − b − b ,T (¯4 , ¯1) = 21 b − b − b + 5 b , T (¯1 , ¯3 , ¯1) = b + 2 b − b + 12 b + b − b ,T (4 , ¯1) = 2 b − b + 12 b + 3 b , T (2 , , ¯2) = − b + 42 b − b + 6 b − b − b − b ,T (¯3 , , ¯1) = 3 b , T (2 , , ¯1) = − b − b + 42 b − b + 6 b − b − b ,T (¯1 , ¯1 , ¯3) = − b + 12 b + 9 b + b , T (3 , , ¯1) = b + 30 b − b + 4 b − b + b + b ,T (¯1 , , ¯3) = 9 b − b − b , T (1 , , ¯3) = − b − b + 14 b − b + 2 b − b − b ,T (1 , ¯2 , 2) = b − b + 6 b − b , T (1 , , ¯2) = 4 b + 90 b − b + 16 b − b + 3 b + 4 b ,T (1 , ¯1 , ¯3) = 4 b − b + 28 b − b , T (¯2 , ¯1 , ¯2) = − b + 6 b + 12 b + 12 b − b − b ,T (1 , ¯2 , ¯2) = 30 b − b − b − b , T (¯2 , , ¯1) = b + 3 b − b + 12 b − b ,T (¯2 , , ¯2) = 12 b − b − b − b , T (¯1 , , , ¯2) = − b + 14 b − b + 6 b + 2 b ,T (2 , ¯1 , 2) = − b + 6 b + 2 b , T (¯1 , , , ¯1) = 180 b − b + 7 b − b − b , (¯1 , ¯2 , , ¯1) = 60 b − b + 3 b − b , T (¯1 , ¯1 , , ¯1) = − b + 84 b − b + 24 b + 2 b ,T (1 , , , ¯2) = 2 b − b − b , T (¯2 , ¯1 , , ¯1) = 150 b + 13 b − b − b + b + 2 b + b ,T (1 , , , ¯1) = − b + 3 b , T (2 , ¯1 , ¯1 , ¯1) = 36 b − b − b + 168 b − b − b − b . All the other 51 weight 5 AMTVs can be obtained by duality Theorem 2.2.1, for e.g., T (1 , ¯1 , ¯1 , ¯2) = b , T (1 , , ¯1 , 2) = b , T (¯1 , , 3) = b , T (¯2 , , 2) = b ,T (¯1 , ¯1 , 3) = T (1 , ¯2 , ¯2) , T (¯1 , ¯2 , 2) = T (2 , ¯1 , ¯2) , T (1 , ¯2 , ¯1 , ¯1) = T (¯2 , ¯1 , , ¯1) ,T (1 , ¯1 , ¯2 , ¯1) = T (¯1 , ¯2 , , ¯1) , T (¯1 , ¯1 , , ¯2) = T (¯1 , , , ¯1) , T (¯1 , , 2) = T (2 , ¯2 , ¯1) . This completes the proof of Theorem 5.0.1.We remark that the integral structure in Theorem 5.0.1 is unlikely to hold in all higherweights. For the corresponding structure for MZVs, we know this integrality breaks down atweight 6 and 7 (see [5, Sec. 2]).We end the paper by the following conjecture on the structure of AMVTs. Conjecture 5.0.2. Let AMTV n be the Q -vector space generated by all AMVTs of weight n . Set AMTV = 1 . Then ∞ X n =0 (dim Q AMTV n ) t n = 11 − t − t − t . Namely, the dimensions form the tribonacci sequence { d w } w ≥ = { , , , , , , . . . } , seeA000073 at oeis.org. We verified the conjecture rigorously when the weight is less than 6 assuming Grothendieck’speriod conjecture. In weight 6, we verified dim Q AMTV = 24 numerically by using PSLQ [2].By duality relations in Theorem 2.2.1 we only need to express 160 AMTVs in terms of theDeligne basis (5.1). Since we do not have double shuffle structure, to cut down computationtime we can produce some additional linear relations such as the lifted relations. We can obtainthese by multiplying every AMTV of weight k ( k = 2 , 3) on each duality relations in weight6 − k . After this step, we reduce the number of AMTVs to be computed to 136. We can furtherreduce this number to 131 if we use the following result: for all ( k j , σ j ) = (1 , σ T ( k , k , k ; σ , σ , σ ) − T ( k , k ; σ , σ ) T ( k ; σ )= σ T ( k , k , k ; σ , σ , σ ) − T ( k , k ; σ , σ ) T ( k ; σ ) , which follows immediately from the [24, (4.5)].Moreover, for all n ≤ AMTV n : TB ′ n = ( T ( s , . . . , s m ; ε , . . . , ε m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s , . . . , s m ∈ { , , } ; ε m = − ε j = sign(1 . − s j ) ∀ j < m ) . From Theorem 5.0.1 we see that TB = TB ′ , TB = TB ′ but we have to make the followingadjustments for larger weights: TB : replace T (¯3) by T (3); TB : replace T (¯2 , ¯2) by T (2 , ¯2); TB : replace T (¯2 , ¯3) by T (2 , ¯3), and T (1 , ¯2 , ¯2) by T (1 , , ¯2);31 B : replace T (¯3 , ¯3) by T (3 , ¯3) . So far, we have not been able to find an apparent pattern of the bases which holds for all theabove cases. Acknowledgments. 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