aa r X i v : . [ m a t h . N T ] J un Duality of Weighted Sum Formulas of AlternatingMultiple T -Values Ce Xu ∗ ∗ School of Mathematics and Statistics, Anhui Normal University,Wuhu 241000, P.R. China
Dedicated to professor Masanobu Kaneko on the occasion of his 60th birthday
Abstract
Recently, a new kind of multiple zeta value level two T ( k ) (which is called multiple T -values) was introduced and studied by Kaneko and Tsumura. In this paper, we define akind of alternating version of multiple T -values, and study several duality formulas of weightedsum formulas about alternating multiple T -values by using the methods of iterated integralrepresentations and series representations. Some special values of alternating multiple T -valuescan also be obtained. Keywords : Kaneko-Tsumura multiple T -values; Alternating multiple T -values; Weighted sumformulas; Duality. AMS Subject Classifications (2020):
We begin with some basic notations. A finite sequence k := ( k , . . . , k r ) of positive integers iscalled an index. We put | k | := k + · · · + k r , d ( k ) := r, and call them the weight and the depth of k , respectively. If k r > k is called admissible. Let I ( k, r ) be the set of all indices of weight k and depth r .For an admissible index k = ( k , . . . , k r ), the multiple zeta values (abbr. MZVs) are definedby ζ ( k , k , . . . , k r ) := X 1o far, surprisingly little work has been done on several variants of multiple zeta values. Inrecent papers [11, 12], Kaneko and Tsumura introduced and studied a new kind of multiple zetavalues of level two T ( k , k , . . . , k r ) : = 2 r X 16. In particular, the author andZhao [21] shown that Z ( j, p ) = ( − p − j π p − j (2 p − j + 1)! . (1.10)The purpose of present paper is to evaluate the duality of products of weighted sum W ( k, r )and Z ( j, p ) by using the method of iterated integrals. Our main results are following threetheorems. Theorem 1.1 For positive integers m and p , we have ( − m p X j =1 W (2 j + 2 m − , m ) Z ( j, p ) = ( − p m X j =1 W (2 j + 2 p − , p ) Z ( j, m ) . (1.11) Theorem 1.2 For positive integers m and p , we have ( − m p X j =1 W (2 j + 2 m − , m − Z ( j, p ) = ( − p m X j =1 W (2 j + 2 p − , p − Z ( j, m ) . (1.12) Theorem 1.3 For positive integers k, m and r , we have W m ( k + r − , r ) + ( − r W k ( m + r − , r )= r − X j =1 ( − j − T ( { } k − , ¯1 , { } j − , ¯1) T ( { } m − , ¯1 , { } r − j − , ¯1) , (1.13) where ( { } − , ¯1) := ∅ , and { l } m denotes the sequence l, . . . , l | {z } m times . Clearly, from Theorems 1.1, 1.2 and 1.3, many explicit relations about weighted sums ofalternating MTVs can be established. For example, by straightforward calculations, we obtain W (5 , 4) + W (5 , 2) = 74 ζ (2) ζ (3) and W (7 , 5) = π ζ (2) W (5 , . Preliminaries In this section we will establish several fundamental formulas for weighed sums W ( k, r ).Kaneko and Tsumura [11,12] introduced the following a new kind of multiple polylogarithmfunction A( k , k , . . . , k r ; z ) := 2 r X ≤ n < ··· 1, we allow z = 1 (In [11], 2 − r A( k , . . . , k r ; z ) isdenoted by Ath( k , . . . , k r ; z ).). When r = 1, the polylogarithm of level 2 A( k ; z ) was consideredby Sasaki in [16].From definition, it is easy to find that ddz A( k , · · · , k r − , k r ; z ) = z A( k , · · · , k r − , k r − z ) ( k r ≥ , − z A( k , · · · , k r − ; z ) ( k r = 1) . (2.2)Hence, by (2.2), we deduce the iterated integral expressionA( k , · · · , k r − , k r ; z ) = z Z dtt · · · dtt | {z } k r − dt − t dtt · · · dtt | {z } k r − − dt − t · · · dtt · · · dtt | {z } k − dt − t = 2 r r Y j =1 ( − k j − Γ( k j ) Z D r ( z ) log k − (cid:18) t t (cid:19) · · · log k r − − (cid:18) t r − t r (cid:19) log k r − (cid:18) t r z (cid:19) (1 − t ) · · · (1 − t r − )(1 − t r ) dt · · · dt r , (2.3)where D r ( z ) := { ( t , . . . , t r ) | < t < · · · < t r < z } ( r ∈ N ) . Here, z Z f k ( t ) dtf k − ( t ) dt · · · f ( t ) dt := Z D k ( z ) f k ( t k ) f k − ( t k − ) · · · f ( t ) dt · · · dt k − dt k . Further, setting t j → zt j in (2.3) givesA( k , . . . , k r − , k r ; z )= 2 r z r r Y j =1 ( − k j − Γ( k j ) Z D r (1) log k − (cid:18) t t (cid:19) · · · log k r − − (cid:18) t r − t r (cid:19) log k r − ( t r )(1 − z t ) · · · (1 − z t r − )(1 − z t r ) dt · · · dt r . (2.4)Applying the changes of variables t j → − t r +1 − j t r +1 − j to (2.4) yieldsA( k , . . . , k r − , k r ; z )= 4 r z r r Y j =1 ( − k j − Γ( k j ) Z D r (1) log k − (cid:18) (1 − t r )(1 + t r − )(1 + t r )(1 − t r − ) (cid:19) (1 + t r ) − z (1 − t r ) · · · log k r − − (cid:18) (1 − t )(1 + t )(1 + t )(1 − t ) (cid:19) (1 + t ) − z (1 − t ) log k r − (cid:18) − t t (cid:19) (1 + t ) − z (1 − t ) dt dt · · · dt r . (2.5)As is well-known, we can regard the function A( k , k , . . . , k r ; z ) as a single-valued holomorphicfunction in the simply connected domain C \ { ( −∞ , − ∪ [1 , + ∞ ) } , via the process of iteratedintegration starting with A(1; z ) = Z z dt/ (1 − t ).Next, we use (2.5) to establish four identities involving weighted sum W ( k, r ), alternatingRiemann zeta values and infinite series whose general n -terms is a product of ( − n /n , multiple T -harmonic sum and multiple S -harmonic sum. First, we need to give the definitions of multiple T -harmonic sum and multiple S -harmonic sum. For an index k := ( k , . . . , k r ), let k m − := ( k , k , . . . , k m − ) and k m := ( k , k , . . . , k m ) . For positive integers n , n , . . . , n r and n , if r = 2 m − D n ( n m − ) := { ( n , n , . . . , n m − , n ) | < n ≤ n < · · · ≤ n m − < n m − ≤ n } , ( n ≥ m ) E n ( n m − ) := { ( n , n , . . . , n m − , n ) | ≤ n < n ≤ · · · < n m − ≤ n m − < n } , ( n > m )and if r = 2 m is an even, we define D n ( n m ) := { ( n , n , . . . , n m , n ) | < n ≤ n < · · · ≤ n m − < n m − ≤ n m < n } , ( n > m ) E n ( n m ) := { ( n , n , . . . , n m , n ) | ≤ n < n ≤ · · · < n m − ≤ n m − < n m ≤ n } , ( n > m ) . Definition 2.1 ( [21]) For positive integer m , the multiple T -harmonic sums ( MTHSs for short)and multiple S -harmonic sums ( MSHSs for short) are defined by T n ( k m − ) := X D n ( n m − ) m − ( Q m − j =1 (2 n j − − k j − (2 n j ) k j )(2 n m − − k m − , (2.6) T n ( k m ) := X D n ( n m ) m Q mj =1 (2 n j − − k j − (2 n j ) k j , (2.7) S n ( k m − ) := X E n ( n m − ) m − ( Q m − j =1 (2 n j − ) k j − (2 n j − k j )(2 n m − ) k m − , (2.8) S n ( k m ) := X E n ( n m ) m Q mj =1 (2 n j − ) k j − (2 n j − k j , (2.9) 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 we let T n ( ∅ ) = S n ( ∅ ) := 1 . We call (2.6) and (2.7) are multiple T -harmonicsums, and call (2.8) and (2.9) are multiple S -harmonic sums. Clearly, according to the definitions of A( k , k , . . . , k r ; z ) and MTHSs, by an elementarycalculation, we can find thatA( k m − ; z ) = 2 ∞ X n =1 T n ( k m − )(2 n − k m − z n − (2.10)6nd A( k m ; z ) = 2 ∞ X n =1 T n ( k m − )(2 n ) k m z n . (2.11)From the definitions of alternating MTVs and A( k , k , . . . , k r ; z ), we haveA( k m − ; i ) = i ( − m ¯ T ( k m − ) (2.12)and A( k m ; i ) = ( − m ¯ T ( k m ) , (2.13)where i is imaginary unit. Kaneko and Tsumura [11] gave the following relationA( { } r ; z ) = 1 r ! (A(1; z )) r = ( − r r ! log r (cid:18) − z z (cid:19) . (2.14)It is clear that A (1; i ) = i π T ( { } r ) = ( − r r ! (cid:16) π (cid:17) r . (2.15)Next, in order to state our main results, we need to give a lemma. Lemma 2.1 ( [21, Theorem 2.1]) For positive integers m and n , the following identities hold Z t n − log m (cid:18) − t t (cid:19) dt = 2(2 m )!2 n − m X j =0 ¯ ζ (2 m − j ) T n ( { } j ) , (2.16) Z t n − log m − (cid:18) − t t (cid:19) dt = − m − n − m − X j =0 ¯ ζ (2 m − − j ) T n ( { } j ) − (2 m − n − S n ( { } m − ) , (2.17) Z t n − log m (cid:18) − t t (cid:19) dt = (2 m )! n m − X j =0 ¯ ζ (2 m − − j ) T n ( { } j +1 )+ (2 m )!2 n S n ( { } m ) , (2.18) Z t n − log m − (cid:18) − t t (cid:19) dt = − (2 m − n m − X j =0 ¯ ζ (2 m − − j ) T n ( { } j +1 ) , (2.19) where ¯ ζ (0) should be interpreted as / wherever it occurs. Using the iterated integral identity (2.5) with the help of Lemma 2.1, we can get thefollowing theorem. 7 heorem 2.2 For positive integers m and p , W (2 p + 2 m − , m ) = 2( − m p − X j =0 ¯ ζ (2 p − − j ) ∞ X n =1 T n ( { } m − ) T n ( { } j +1 ) n ( − n , (2.20) W (2 p + 2 m − , m ) = 2( − m p − X j =0 ¯ ζ (2 p − − j ) ∞ X n =1 T n ( { } m − ) T n ( { } j +1 ) n ( − n + ( − m ∞ X n =1 T n ( { } m − ) S n ( { } p − ) n ( − n , (2.21) W (2 p + 2 m − , m − 1) = 4( − m − p − X j =0 ¯ ζ (2 p − − j ) ∞ X n =1 T n ( { } m − ) T n ( { } j )2 n − − n + 2( − m − ∞ X n =1 T n ( { } m − ) S n ( { } p − )2 n − − n , (2.22) W (2 p + 2 m − , m − 1) = 4( − m − p − X j =0 ¯ ζ (2 p − − j ) ∞ X n =1 T n ( { } m − ) T n ( { } j )2 n − − n . (2.23) Proof. By using (2.12) and (2.13) we know that¯ T ( k , k , . . . , k r ) = i r A( k , k , . . . , k r ; i ) . Then summing both sides of above identity and using (2.5), we have W ( k + r − , r ) = ( − k + r − ( k − r Z D r (1) log k − (cid:18) − t r t r (cid:19) (1 + t ) · · · (1 + t r ) dt · · · dt r . (2.24)Expand the (1 + t j ) − into geometric series to deduce that W ( k + r − , r ) = ( − k − ( k − r X 1, then W (2 p + 2 m − , m ) = 2( − m (2 p − ∞ X n =1 ( − n T n ( { } m − ) Z t n − log p − (cid:18) − t t (cid:19) dt, r = 2 m − k = 2 p , then W (2 p + 2 m − , m − 1) = 2( − m (2 p − ∞ X n =1 ( − n T n ( { } m − ) Z t n − log p − (cid:18) − t t (cid:19) dt, if r = 2 m − k = 2 p − 1, then W (2 p + 2 m − , m − 1) = 2( − m − (2 p − ∞ X n =1 ( − n T n ( { } m − ) Z t n − log p − (cid:18) − t t (cid:19) dt. Thus, with the help of formulas (2.16)-(2.19), we may easily deduce these desired evaluations. (cid:3) Next, we give several special cases. Setting m = p = 1 in (2.20) yields ∞ X n =1 T n (1) n ( − n − = ¯ T (1 , 2) + ¯ T (2 , 1) = 74 ζ (3) , (2.25)where we used these two well-known results¯ T (1 , 2) = − ζ (3) + π t (2) and ¯ T (2 , 1) = 72 ζ (3) − π t (2) . Letting m = 1 in (2.23) gives¯ T (2 p − 1) = 2 p − X j =1 ( − j ¯ ζ (2 p − − j ) ¯ T ( { } j +1 ) . (2.26)Now, we end this section by a theorem. Theorem 2.3 For positive integers r and k , ¯ T ( { } r − , k ) = r X j =1 ( − j − ¯ T ( { } r − j ) W ( k + j − , j ) . (2.27) Proof. According to the definitions of alternating MTVs and using (2.5) with z = i, k = · · · = k r − = 1 and k r = 1, we have¯ T ( { } r − , k ) = ( − k + r − ( k − r Z D r (1) log k − (cid:18) − t t (cid:19) (1 + t ) · · · (1 + t r ) dt · · · dt r . (2.28)Note that the two integrals Z In this section, we give the proofs of Theorems 1.1 and 1.2. We need the following a lemma. Lemma 3.1 ( [21, 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 , (3.1) hold, then B p = p X j =1 C j p − j X k =1 ( − k X i
Theorem 3.2 For positive integers m and p , ∞ X n =1 T n ( { } m − ) T n ( { } p − ) n ( − n = ( − m p X j =1 W (2 j + 2 m − , m ) Z ( j, p ) . (3.3) Proof. Setting A j,p = 2¯ ζ (2 p − j ) , B j = ∞ X n =1 T n ( { } m − ) T n ( { } j − ) n ( − n and C p = ( − m W (2 p + 2 m − , m )10n Lemma 3.1 and using (2.20) gives the desired formula. (cid:3) Putting m = 1 and p = 2 in (3.3), we get ∞ X n =1 T n (1) T n (1 , , n ( − n = 74 ζ (2) ζ (3) − W (5 , 2) = 2116 ζ (2) ζ (3) − ζ (5) , where we used the formula (2.25) and the identity W (5 , 2) = 716 ζ (2) ζ (3) + 3116 ζ (5) . Theorem 3.3 For positive integers m and p , ∞ X n =1 T n ( { } m − ) T n ( { } p − )2 n − − n = ( − m − p X j =1 W (2 j + 2 m − , m − Z ( j, p ) . (3.4) Proof. Setting A j,p := 2¯ ζ (2 p − j ) , B j := 2 ∞ X n =1 T n ( { } m − ) T n ( { } j − )2 n − − n and C p := ( − m − W (2 p + 2 m − , m − (cid:3) Letting m = p = 2 in (3.4), we have2 ∞ X n =1 T n (1 , n − − n = − W (5 , 3) + 2¯ ζ (2) ¯ T (1 , , . Applying (2.21) and (2.22), we know that for positive integers m and p , the sums ∞ X n =1 T n ( { } m − ) S n ( { } p − ) n ( − n and ∞ X n =1 T n ( { } m − ) S n ( { } p − )2 n − − n can also be evaluated by weighted sums W ( k, r ) and Z ( j, p ). Proofs of Theorems 1.1 and 1.2 . Changing ( m, p ) to ( p, m ) in (3.3) and (3.4), and usingthe duality of series on the left hand sides, then ∞ X n =1 T n ( { } m − ) T n ( { } p − ) n ( − n = ( − m p X j =1 W (2 j + 2 m − , m ) Z ( j, p )= ( − p m X j =1 W (2 j + 2 p − , p ) Z ( j, m )and 2 ∞ X n =1 T n ( { } m − ) T n ( { } p − )2 n − − n = ( − m − p X j =1 W (2 j + 2 m − , m − Z ( j, p )11 ( − p − m X j =1 W (2 j + 2 p − , p − Z ( j, m ) . Thus, we prove Theorems 1.1 and 1.2. (cid:3) We get some cases. Setting m = 1 in (1.11) and (1.12) yield W (2 p + 1 , p ) = ( − p − p X j =1 W (2 j + 1 , Z ( j, p ) (3.5)and ¯ T ( { } p − ) = ( − p − p X j =1 ¯ T (2 j − Z ( j, p ) . (3.6)Now, we give the explicit expression of weighted sums W (2 k + 1 , 2) via t -values and alter-nating zeta values. Theorem 3.4 For positive integer k , W (2 k + 1 , 2) = 12 k − k X j =1 e t (2 j + 1)¯ ζ (2 k − j ) , (3.7) where e t ( k ) := 2 k t ( k ) = (2 k − ζ ( k ) if k > and e t (1) := 0 if k = 1 .Proof. From [19, Eq.(3.10)], Wang and the author gave the result(1 − ( − p + q ) ∞ X n =1 h ( p ) n n q ( − n − = − ( − p (1 + ( − q ) e t ( p )¯ ζ ( q ) + ( − p (cid:18) p + q − p − (cid:19)e t ( p + q ) − ( − p p − X k =0 (( − k + 1) (cid:18) p + q − k − q − (cid:19) ¯ t ( k + 1)¯ t ( p + q − k − − p [ q/ X j =1 (cid:18) p + q − j − p − (cid:19) ¯ ζ (2 j ) e t ( p + q − j ) . (3.8)According to definition, we have W (2 k + 1 , 2) = X k k k +1 ,k ,k ≥ k + k − ∞ X n =1 h ( k ) n n k ( − n − . Thus, applying (3.8) with p = k and q = k , by an elementary calculation, we complete thisproof. (cid:3) Hence, using (3.5) and (3.7), we obtain the following description. Corollary 3.5 For positive integer p , the weighted sums W (2 p +1 , p ) can be expressed in termsof (alternating) Riemann zeta values. As two examples, for p = 1 and 2, we have W (3 , 2) = 74 ζ (3) and W (5 , 4) = 2116 ζ (2) ζ (3) − ζ (5) . Proof of Theorem 1.3 In this section we prove Theorem 1.3. First, we need to give a lemma. Lemma 4.1 ( [20, Lemma 2.5]) If f i ( i = 1 , . . . , m ) are integrable real functions, the followingidentity holds: g ( f , f , · · · , f m ) + ( − m g ( f m , f m − , · · · , f )= m − X i =1 ( − i − g ( f i , f i − , · · · , f ) g ( f i +1 , f i +2 · · · , f m ) , (4.1) where g ( f , f , · · · , f m ) is defined by g ( f , f , · · · , f m ) := Z For positive integers k and r , W ( k + r − , r ) + ( − r ¯ T ( { } r − , k ) = r − X j =1 ( − j − T ( { } k − , ¯1 , { } j − , ¯1) ¯ T ( { } r − j ) . (4.4)Setting r = 2 in (4.4) yields W ( k + 1 , 2) + ¯ T (1 , k ) = T ( { } k − , ¯1 , ¯1) ¯ T (1) . (4.5)where ¯ T (1) = − π/ heorem 4.3 For integers p ≥ and r ≥ , T ( { } p − , ¯1 , { } r − , ¯1) = ¯ T ( { } r − , p + 1) , (4.6) where ( { } − , ¯1) := ∅ .Proof. In (4.3), applying the changes of variables t j → − t r +1 − j t r +1 − j to (2.4) gives( − p + r p !2 r T ( { } p − , ¯1 , { } r − , ¯1) = Z D r (1) log p ( t r )(1 + t ) · · · (1 + t r ) dt · · · dt r = ( − p + r p !2 r ¯ T ( { } r − , p + 1) , where in the last step, we expanded the (1 + t j ) − in geometric series and used the followingwell-known identity Z t n − log p ( t ) dt = p ! ( − p n p +1 . Thus, we complete this proof. (cid:3) Letting r = 1 in (4.6) yields T ( { } p − , ¯1 , ¯1) = − p ¯ t ( p + 1) . (4.7)In particular, if k = 2, then we have T (¯1 , ¯1) = − 12 ¯ t (2) = − G, where G := ∞ X n =1 ( − n − / (2 n − is Catalan’s constant (see [13, Eq. (7.4.4.183a)]). Hence,substituting (4.7) into (4.5) with p = k − 1, we get the following corollary. Corollary 4.4 For positive integer k , W ( k + 1 , 2) = π k ¯ t ( k ) − ¯ T (1 , k ) . (4.8) Remark 4.5 The formula (4.8) can be obtained by Theorem 2.3 with r = 2 . Further, applying (4.6) with p = k − 1, the formula (1.13) can be rewritten in the form W m ( k + r − , r ) + ( − r W k ( m + r − , r ) = r − X j =1 ( − j − ¯ T ( { } j − , k ) ¯ T ( { } r − j − , m ) (4.9)and the formula (4.4) can be rewritten in the form W ( k + r − , r ) + ( − r ¯ T ( { } r − , k ) = r − X j =1 ( − j − ¯ T ( { } j − , k ) ¯ T ( { } r − j ) . (4.10)Clearly, (4.10) can also be immediately obtained from (4.9) with m = 1. Moreover, setting r = 2in (4.10) yields (4.8).Now, we end this section by a theorem. 14 heorem 4.6 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 − z )+ ( − r X k ··· + kr = m + r − ,k ,...,kr ≥ (cid:18) k r + k − k − (cid:19) A( k , . . . , k r − , k r + k − z )= r − X j =1 ( − j − A( { } r − − j , m ; z )A( { } j − , k ; z ) . (4.11) Proof. 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 − z )= ( − k + m ( m − k − r z r Z D r (1) log m − (cid:18) − t t (cid:19) log k − (cid:18) − t r t r (cid:19) dt · · · dt r [(1 + t ) − z (1 − t ) ] · · · [(1 + t r ) − z (1 − t r ) ] . (4.12)Letting k = 1 yieldsA( { } r − , m ; z ) = ( − m − ( m − r z r Z D r (1) log m − (cid:18) − t t (cid:19) dt · · · dt r [(1 + t ) − z (1 − t ) ] · · · [(1 + t r ) − z (1 − t r ) ] . 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