Quadratic relations for a q-analogue of multiple zeta values
aa r X i v : . [ m a t h . N T ] A ug QUADRATIC RELATIONS FORA q -ANALOGUE OF MULTIPLE ZETA VALUES YOSHIHIRO TAKEYAMA
Abstract.
We obtain a class of quadratic relations for a q -analogue of multiplezeta values ( q MZV’s). In the limit q →
1, it turns into Kawashima’s relation formultiple zeta values. As a corollary we find that q MZV’s satisfy the linear relationcontained in Kawashima’s relation. In the proof we make use of a q -analogue ofNewton series and Bradley’s duality formula for finite multiple harmonic q -series. Introduction
In this paper we prove quadratic relations for a q -analogue of multiple zeta values( q MZV’s, for short). The relations are of a similar form to Kawashima’s relation formultiple zeta values (MZV’s).First let us recall the definition of q MZV [1, 11]. Let k = ( k , . . . , k r ) be an r -tupleof positive integers such that k ≥
2. Then q MZV ζ q ( k ) is a q -series defined by ζ q ( k ) := X m > ··· >m r > q ( k − m + ··· +( k r − m r [ m ] k · · · [ m r ] k r , (1.1)where [ n ] is the q -integer [ n ] := 1 − q n − q . (1.2)Since k ≥
2, the right hand side of (1.1) is well-defined as a formal power series of q . If we regard q as a complex variable, it is absolutely convergent in | q | <
1. In thelimit as q → q MZV turns into MZV defined by ζ ( k ) := X m > ··· >m r > m k · · · m k r r . An interesting point is that q MZV’s satisfy many relations in the same form as thosefor MZV’s. For example, q MZV’s satisfy Ohno’s relation [6], the cyclic sum formula[3, 7] and Ohno-Zagier’s relation [8]. See [1] for the proof of Ohno’s relation and thecyclic sum formula for q MZV’s, and see [9] for that of Ohno-Zagier’s relation.In this paper we prove a class of quadratic relations for q MZV’s (see Theorem4.6 below). In the limit as q →
1, it turns into Kawashima’s relation [4] for MZV’s.Some of our relations are linear, and they are completely the same as the linearpart of Kawashima’s relation (Corollary 4.7). It is known that the linear part of
Kawashima’s relation contains Ohno’s relation [4] and the cyclic sum formula [10],and hence we find again that q MZV’s also satisfy them.The proof of our quadratic relations proceeds in a similar manner to that ofKawashima’s relation. The ingredients are a q -analogue of Newton series and finitemultiple harmonic q -series. Let b = { b ( n ) } ∞ n =0 be a sequence of formal power seriesin q . Then we define a sequence ∇ q ( b ) by ∇ q ( b )( n ) := n X i =0 q i ( q − n ) i ( q ) i b ( i )and consider the series f ∇ q ( b ) ( z ) := ∞ X n =0 ∇ q ( b )( n ) z n ( z − ) n ( q ) n , where ( x ) n is the q -shifted factorial( x ) n := n − Y j =0 (1 − xq j ) . (1.3)Under some condition for ∇ q ( b ), the series f ∇ q ( b ) ( z ) is well-defined as an element of Q [[ q, z ]] and it satisfies f ∇ q ( b ) ( q m ) = b ( m ) for m ≥ f ∇ q ( b ) ( z ) interpolates the sequence b , and can be regarded as a q -analogue ofNewton series. It has a nice property: f ∇ q ( b ) f ∇ q ( b ) = f ∇ q ( b b ) . (1.4)Now consider the finite multiple harmonic q -series S k ( n ) defined by S k ( n ) := X n ≥ m ≥···≥ m r ≥ q m + m + ··· + m r [ m ] k · · · [ m r ] k r . Any product of S k ’s can be written as a linear combination of them with coeffi-cients in Q [(1 − q )]. The duality formula due to Bradley [2] (see Proposition 2.1below) implies that the coefficients are q MZV’s in the expansion of f ∇ q ( S k ) at z = 1.Therefore, by expanding f ∇ q ( S k ) f ∇ q ( S k ′ ) = f ∇ q ( S k S k ′ ) at z = 1, we obtain quadraticrelations for q MZV’s.The paper is organized as follows. In Section 2 we define finite multiple harmonic q -series and describe their algebraic structure by making use of a non-commutativepolynomial ring. In Section 3 we define a q -analogue of Newton series and provethe key relation (1.4). In Section 4 we prove the quadratic relations and see that q MZV’s satisfy the linear part of Kawashima’s relation for MZV’s.In this paper we denote by N the set of non-negative integers. UADRATIC RELATIONS FOR q MZV 3 Algebraic structure of finite multiple harmonic q -series Finite multiple harmonic q -series. Let ~ be a formal variable and C := Q [ ~ ]the coefficient ring. Denote by h the non-commutative polynomial algebra over C freely generated by the set of alphabets { z n } ∞ n =1 . We define the depth of a word u = z i · · · z i r by dep( u ) := r .Let R := Q [[ q ]] be the ring of formal power series in q . We endow R with C -module structure such that ~ acts as multiplication by 1 − q . Denote by R N the setof sequences b = { b ( n ) } ∞ n =0 of formal power series b ( n ) ∈ R . Then R N is a C -algebrawith the product defined by ( bc )( n ) := b ( n ) c ( n ) for b, c ∈ R N .For a word u = z k · · · z k r , we define S u , A u , A ⋆u ∈ R N by S u ( n ) := X n ≥ m ≥···≥ m r ≥ q m + m + ··· + m r [ m ] k · · · [ m r ] k r , (2.1) A u ( n ) := X n ≥ m > ··· >m r > q ( k − m +( k − m + ··· +( k r − m r [ m ] k · · · [ m r ] k r , (2.2) A ⋆u ( n ) := X n ≥ m ≥···≥ m r ≥ q ( k − m +( k − m + ··· +( k r − m r [ m ] k · · · [ m r ] k r , (2.3)where [ n ] is the q -integer defined by (1.2). Setting S ( n ) , A ( n ) , A ⋆ ( n ) ≡
1, weextend the correspondence u S u , A u , A ⋆u to the C -linear map S, A, A ⋆ : h → R N .Let h > be the C -submodule consisting of non-constant elements, that is, h > = P ∞ k =1 P i ,...,i k ≥ C z i · · · z i k . For a word u = z k · · · z k r ∈ h > , set s u , a u ∈ R N by s u ( n ) := X n +1= m ≥ m ≥···≥ m r ≥ q k m +( k − m + ··· +( k r − m r [ m ] k · · · [ m r ] k r , (2.4) a u ( n ) := X n +1= m >m > ··· >m r > q ( k − m +( k − m + ··· +( k r − m r [ m ] k · · · [ m r ] k r . (2.5)Extending by C -linearity we define two maps s, a : h > → R N . Note that if w = z i w ′ ( w, w ′ ∈ h , i ≥
1) we have s w ( n ) = q i ( n +1) [ n + 1] i A ⋆w ′ ( n + 1) , a w ( n ) = q ( i − n +1) [ n + 1] i A w ′ ( n ) . (2.6)We define a map φ : h → h as follows. For a word u = z k · · · z k r , consider the set I u = { P ji =1 k i | ≤ j < r } . Denote by I cu the set consisting of positive integers whichare less than or equal to P ri =1 k i not belonging to I u . Set I cu = { p , . . . , p l } ( p < · · · < p l ) and define φ ( u ) := z k ′ · · · z k ′ l , where k ′ = p , k ′ i = p i − p i − (2 ≤ i ≤ l ).Extending by C -linearity φ is defined as a map on h . Note that φ = id. YOSHIHIRO TAKEYAMA
Now we define a C -linear map ∇ q : R N → R N by ∇ q ( b )( n ) := n X i =0 q i ( q − n ) i ( q ) i b ( i ) , where ( x ) n is the q -shifted factorial defined by (1.3). The following duality formulais due to Bradley [2] (see also [5]): Proposition 2.1.
For w ∈ h > , we have ∇ q ( S w )( n ) = (cid:26) n = 0) , − s φ ( w ) ( n −
1) ( n ≥ . Remark 2.2.
The operator ∇ q has another description. Consider the differenceoperator ∆ t : R N → R N defined by ∆ t ( b )( n ) := b ( n ) − tb ( n + 1). Then we have ∇ q ( b )( n ) = (∆ q − ( n − ◦ · · · ◦ ∆ q − ◦ ∆ ( b ))(0) (see Corollary 2.7 in [5]).2.2. A q -analogue of multiple zeta (star) values. Introduce the valuation v : R → Z ≥ ∪ { + ∞} defined by v ( f ) := inf { j | c j = 0 } for f = P ∞ j =0 c j q j , and endow R with the topology such that the sets { f + φ | v ( φ ) ≥ n } ( n = 0 , , . . . ) formneighborhood base at f ∈ R . Then R is complete.Denote by h the C -submodule of h generated by 1 and the words z k · · · z k r satisfying r ≥ k ≥
2. If w ∈ h , the limit lim n →∞ A w ( n ) = P ∞ k =0 a w ( k )converges in R since v ( a w ( n )) ≥ n + 1. We call it a q -analogue of multiple zeta value ( q MZV) and denote it by ζ q ( w ). If w is a word w = z k · · · z k r ∈ h , it is given by ζ q ( z k · · · z k r ) := X m > ··· >m r > q ( k − m +( k − m + ··· +( k r − m r [ m ] k · · · [ m r ] k r . We also define a q -analogue of multiple zeta star value ( q MZSV) (or that of non-strictmultiple zeta value ) by ζ ⋆q ( z k · · · z k r ) := X m ≥···≥ m r ≥ q ( k − m +( k − m + ··· +( k r − m r [ m ] k · · · [ m r ] k r . If we regard q as a complex variable, q MZV and q MZSV are absolutely convergentin | q | <
1. Therefore in fact they are convergent series. In the limit q →
1, theyconverge to multiple zeta values (MZV) and multiple zeta star values (or non-strictmultiple zeta values) defined by ζ ( z k · · · z k r ) := X m > ··· >m r > m k · · · m k r r and ζ ⋆ ( z k · · · z k r ) := X m ≥···≥ m r ≥ m k · · · m k r r , respectively. UADRATIC RELATIONS FOR q MZV 5
Algebraic structure.
Let z be the C -submodule of h generated by { z i } ∞ i =1 .We define three products ◦ , ◦ ± on z by setting z i ◦ z j := z i + j , z i ◦ ± z j := ± z i + j + ~ z i + j − and extending by C -linearity. These products are associative and commutative.Define two C -bilinear products ∗ ± on h inductively by1 ∗ ± w = w, w ∗ ± w, ( z i w ) ∗ ± ( z j w ) = z i ( w ∗ ± z j w ) + z j ( z i w ∗ ± w ) + ( z i ◦ ± z j )( w ∗ ± w )(2.7)for i, j ≥ w, w , w ∈ h . These products are commutative. Proposition 2.3.
Let w , w ∈ h . Then S w S w = S w ∗ − w and A w A w = A w ∗ + w .Proof. It suffices to show that S w S w and A w A w follow the recurrence relation(2.7) for ∗ − and ∗ + , respectively. It can be checked by using q m [ m ] i + j = q m [ m ] i + j − (1 − q ) q m [ m ] i + j − and q ( i + j − m [ m ] i + j = q ( i + j − m [ m ] i + j + (1 − q ) q ( i + j − m [ m ] i + j − (2.8)for i, j ≥ (cid:3) The products ◦ and ◦ + determine z -module structures on h , which we denote bythe same letters ◦ and ◦ + , such that z i ◦ , z i ◦ ( z j w ) := ( z i ◦ z j ) w ( i, j ≥ , w ∈ h )and the above formulas where ◦ is replaced with ◦ ± . By convention we set z ◦ c = 0for c ∈ C and z ◦ w = w for w ∈ h > . Then z i ◦ + w = ( z i + ~ z i − ) ◦ w for i ≥ w ∈ h .Consider the C -linear map d q on h defined inductively by d q (1) = 1 , d q ( z i w ) = z i d q ( w ) + z i ◦ + d q ( w ) ( i ≥ , w ∈ h ) . Note that d q is invertible and its inverse satisfies d − q (1) = 1 and d − q ( z i w ) = z i d − q ( w ) − z i ◦ + d − q ( w ) ( i ≥ , w ∈ h ) . Lemma 2.4.
For i ≥ and w ∈ h , we have d q ( z i ◦ w ) = z i ◦ d q ( w ) , that is, themap d q commutes with the action ◦ of z .Proof. For w = 1 it is trivial. Suppose that w is a word and set w = z j w ′ . Thenwe have d q ( z i ◦ w ) = d q ( z i + j w ′ ) = z i + j d q ( w ′ ) + z i + j ◦ + d q ( w ′ ) = z i ◦ ( z j d q ( w ′ )) +( z i ◦ z j ) ◦ + d q ( w ′ ). Using ( z i ◦ z j ) ◦ + z k = z i ◦ ( z j ◦ + z k ) for i, j, k ≥
1, we see that( z i ◦ z j ) ◦ + d q ( w ′ ) = z i ◦ ( z j ◦ + d q ( w ′ )). Thus we get d q ( z i ◦ w ) = z i ◦ d q ( w ) for a word w . From the C -linearity of d q , we obtain the lemma. (cid:3) Proposition 2.5.
Let w ∈ h . Then A ⋆w = A d q ( w ) . YOSHIHIRO TAKEYAMA
Proof.
It is enough to prove the case where w is a word. We prove the propositionby induction on the depth of w . If dep( w ) = 1, that is, w = z i for some i ≥
1, wehave A ⋆z i = A z i = A d q ( z i ) .From the definition of A and A ⋆ we find that n X m =1 q ( i − m [ m ] i A ⋆w ( m ) = A ⋆z i w ( n ) , n X m =1 q ( i − m [ m ] i A w ( m ) = A z i w + z i ◦ + w ( n )for w ∈ h . To show the second formula, divide the sum in the definition (2.2) of A w ( m ) into two parts with m = m and with m < m , and use (2.8) for the firstpart. Now suppose that dep( w ) > w ). Then setting w = z i w ′ , we find A ⋆w ( n ) = n X m =1 q ( i − m [ m ] i A ⋆w ′ ( m ) = n X m =1 q ( i − m [ m ] i A d q ( w ′ ) ( m )= A z i d q ( w ′ )+ z i ◦ + d q ( w ′ ) ( n ) = A d q ( w ) ( n ) . (cid:3) Corollary 2.6.
For w ∈ h , we have ζ ⋆q ( w ) = ζ q ( d q ( w )) . We define a C -bilinear product ⊛ q on h > by( z i w ) ⊛ q ( z j w ) := ( z i ◦ z j )( w ∗ + w ) ( i, j ≥ , w , w ∈ h ) . (2.9)It is commutative and associative. Proposition 2.7.
Let w , w ∈ h > . Then s w a w = a d q ( w ) ⊛ q w . To prove Proposition 2.7 we need the following lemma:
Lemma 2.8.
For n ∈ N and w ∈ h , we have q ( i − n +1) [ n + 1] i A w ( n + 1) = a z i w + z i ◦ + w ( n ) . (2.10) Proof.
It is trivial for w = 1. Without loss of generality we can assume that w is aword. Set w = z j w ′ . Then we have A w ( n + 1) = q ( j − n +1) [ n + 1] j A w ′ ( n ) + A w ( n ) . Substitute it into the left hand side of (2.10) and use (2.8). Then we obtain (cid:18) q ( i + j − n +1) [ n + 1] i + j + (1 − q ) q ( i + j − n +1) [ n + 1] i + j − (cid:19) A w ′ ( n ) + q ( i − n +1) [ n + 1] i A w ′ ( n ) . Using (2.6) again, we see that it is equal to the right hand side of (2.10). (cid:3)
UADRATIC RELATIONS FOR q MZV 7
Proof of Proposition 2.7.
We can assume that w and w are words. Set w = z i w ′ and w = z j w ′ . Using (2.6) and Proposition 2.5, we have( s w a w )( n ) = q ( i + j − n +1) [ n + 1] i + j A d q ( w ′ ) ( n + 1) A w ′ ( n ) . It is equal to a d q ( z i + j w ′ ) ( n ) A w ′ ( n ) because of Lemma 2.8 and the definition of d q .Now introduce a C -bilinear map △ : h > × h → h > uniquely determined from theproperty ( z i w ′ ) △ w ′′ := z i ( w ′ ∗ + w ′′ ) ( i ≥ . Then a w A w ′′ = a w △ w ′′ for w ∈ h > and w ′′ ∈ h because of (2.6) and Proposition2.3. Therefore Proposition 2.7 is reduced to the identity( d q ( z i + j w ′ )) △ w ′ = ( d q ( z i w ′ )) ⊛ q ( z j w ′ )(2.11)for i, j ≥ w ′ , w ′ ∈ h .Let us prove (2.11). From the definition of d q and △ , the left hand side is equalto z i + j ( d q ( w ′ ) ∗ + w ′ ) + ( z i + j ◦ + d q ( w ′ )) △ w ′ . The first term is equal to ( z i d q ( w ′ )) ⊛ q ( z j w ′ ) because z i + j = z i ◦ z j . Now note that z i + j ◦ + z k = ( z i ◦ + z k ) ◦ z j for any i, j, k ≥
1. Using this we see that the second termis equal to ( z i ◦ + d q ( w ′ )) ⊛ q ( z j w ′ ). Summing up these two terms we get the righthand side of (2.11). (cid:3) A q -analogue of Newton series We consider the ring of formal power series R [[ z ]] = Q [[ q, z ]]. Introduce thevaluation on R [[ z ]] by v ( f ) = inf { i + j | c ij = 0 } for f = P ∞ i,j =0 c ij q i z j and endow R [[ z ]] with the structure of topological Q -algebra.Define polynomials B n ( z ) ( n ∈ N ) by B ( z ) := 1 , B n ( z ) := z n ( z − ) n ( q ) n = n Y j =1 z − q j − − q j ( n ≥ . Proposition 3.1.
Let b ∈ R N and m, l ∈ N . Then m X n =0 ∇ ( b )( l + n ) B n ( q m ) = q − lm l X j =0 q j ( q − l ) j ( q ) j b ( j + m ) . (3.1) In particular, we have P mn =0 ∇ ( b )( n ) B n ( q m ) = b ( m ) . In the proof of Proposition 3.1 we use the following formula:
YOSHIHIRO TAKEYAMA
Lemma 3.2.
For m, j ∈ N we have q ) j m X n =0 q mn ( q − m ) n ( tq − n ) j ( q ) n = ≤ j < m ) , ( q − t ) m ( t ) j − m ( q ) j − m ( m ≤ j ) . Proof.
Multiply the both sides by s j and sum up over j ∈ N using the q -binomialformula ∞ X n =0 ( a ) n ( q ) n x n = ( ax ) ∞ ( x ) ∞ . (3.2)Then we see that the identity to prove is equivalent to m X n =0 q mn ( q − m ) n ( q ) n ( q − n st ) n = ( q − st ) m . (3.3)Using ( q − a x ) n = ( − x ) n q − an + n ( n − / ( q a − n +1 x − ) n , (3.4)we find that the left hand side of (3.3) is equal to( q ) m m X n =0 ( qs − t − ) n ( q ) n ( q ) m − n ( q − st ) n . It is equal to the coefficient of x m in( q ) m ∞ X i =0 ( q − stx ) i ( qs − t − ) i ( q ) i ! ∞ X i =0 x i ( q ) i ! , and the above product is equal to( q ) m ( x ) ∞ ( q − stx ) ∞ x ) ∞ = ( q ) m q − stx ) ∞ = ( q ) m ∞ X j =0 ( q − stx ) j ( q ) j from the q -binomial formula (3.2). Thus we obtain the right hand side of (3.3). (cid:3) Proof of Proposition 3.1.
From ( q − l − n ) j = 0 for j > l + n , we have m X n =0 ∇ ( b )( l + n ) B n ( q m ) = l + m X j =0 q j ( q ) j b ( j ) m X n =0 q mn ( q − m ) n ( q − l − n ) j ( q ) n . The second sum in the right hand side can be calculated by using Lemma 3.2 with t = q − l . Then we obtain the right hand side of (3.1). (cid:3) Suppose that c ∈ R N satisfies v ( c ( n )) ≥ n for all n ∈ N . (3.5) UADRATIC RELATIONS FOR q MZV 9
Set f c ( z ) := ∞ X n =0 c ( n ) B n ( z ) . (3.6)Then the series f c ( z ) converges in R [[ z ]]. If c = ∇ q ( b ) satisfies the condition (3.5),the series f ∇ q ( b ) is well-defined and we have f ∇ q ( b ) ( q m ) = b ( m ) for all m ∈ N fromLemma 3.1. Thus we may regard the series f ∇ q ( b ) ( z ) as a q -analogue of Newtonseries interpolating b ∈ R N .Let us prove two properties of the series f c . Proposition 3.3.
Suppose that c ∈ R N satisfies the condition (3.5) . Then thefollowing equality holds in R [[ z ]] : f c ( z ) = c (0) + ∞ X m =1 ∞ X n =1 c ( n ) a z m ( n − ! (cid:18) z − − q (cid:19) m , where a z m ( n ) is defined by (2.5) .Proof. It follows from B n ( z ) = 1[ n ] z − − q n − Y j =1 (cid:18) j ] z − − q (cid:19) = n X m =1 a z m ( n − (cid:18) z − − q (cid:19) m and a z m ( k −
1) = 0 for k < m . (cid:3) Proposition 3.4.
Suppose that ∇ q ( b i ) ∈ R N ( i = 1 , satisfy the condition (3.5) .Then we have f ∇ q ( b ) ( z ) f ∇ q ( b ) ( z ) = f ∇ q ( b b ) ( z ) . We prove two lemmas to show Proposition 3.4.
Lemma 3.5.
For n ∈ N we have B n ( z ) = y n n X j =0 ( y − ) n − j ( q ) n − j B j ( y − z ) . (3.7) Proof.
The right hand side is equal to the coefficient of t n in ∞ X j =0 ( ty ) j ( y − ) j ( q ) j ! ∞ X j =0 ( tz ) j ( yz − ) j ( q ) j ! . From the q -binomial formula (3.2), the above product is equal to( t ) ∞ ( yt ) ∞ ( yt ) ∞ ( tz ) ∞ = ( t ) ∞ ( tz ) ∞ = ∞ X n =0 t n B n ( z ) . This completes the proof. (cid:3)
Lemma 3.6.
Suppose that c , c ∈ R N satisfy the condition (3.5) . Set c ∈ R N by c ( n ) := n X k =0 (cid:20) nk (cid:21) q c ( k ) k X j =0 c ( n − k + j ) B j ( q k ) , where (cid:2) nk (cid:3) q is the q -binomial coefficient (cid:20) nk (cid:21) q := ( q ) n ( q ) k ( q ) n − k . Then we have f c ( z ) f c ( z ) = f c ( z ) .Proof. Substitute (3.7) into f c ( z ) f c ( z ) = P ∞ m,n =0 c ( m ) c ( n ) B m ( z ) B n ( z ) setting y = q m . Using B m ( z ) B j ( q − m z ) = q − jm B m + j ( z ), we find that f c ( z ) f c ( z ) = ∞ X m,n =0 n X j =0 c ( m ) c ( n ) q ( n − j ) m ( q − m ) n − j ( q ) n − j (cid:20) m + jm (cid:21) q B m + j ( z ) . Then we can see that it is equal to f c ( z ). (cid:3) Proof of Proposition 3.4.
From Proposition 3.1 and Lemma 3.6, it suffices to provethat X ≤ i ≤ j ≤ n q i + j ( q ) i b ( i ) b ( j ) j X k = i q k ( k − n − (cid:20) nk (cid:21) q ( q − k ) i ( q − n + k ) j − k ( q ) j − k = ∇ ( b b )( n ) . (3.8)Using (3.4), we see that the second sum in the left hand side above is equal to( − j q j ( j − / − nj − i ( q ) n ( q ) n − j ( q ) j − i j − i X k =0 ( − k q k ( k − / (cid:20) j − ik (cid:21) q . For N ∈ N , we have N X k =0 ( − k q k ( k − / (cid:20) Nk (cid:21) q = δ N, . Thus we find that the left hand side of (3.8) is equal to n X i =0 ( − i q − ni + i ( i +1) / ( q ) i ( q ) n ( q ) n − i b ( i ) b ( i ) = n X i =0 q i ( q − n ) i ( q ) i ( b b )( i ) = ∇ ( b b )( n ) . (cid:3) UADRATIC RELATIONS FOR q MZV 11 A q -analogue of Kawashima’s relation Kawashima’s relation.
Define a product ∗ on h inductively by1 ∗ w = w, w ∗ w, ( z i w ) ∗ ( z j w ) = z i ( w ∗ z j w ) + z j ( z i w ∗ w ) − z i + j ( w ∗ w )for i, j ≥ w, w , w ∈ h . We also define a product ⊛ on h > by the formula(2.9) where the product ∗ + in the right hand side is replaced with ∗ . We define a C -linear map d : h → h by the properties d (1) = 1 and d ( z i w ) = z i d ( w ) + z i ◦ d ( w )for w ∈ h . The products ∗ , ⊛ and the map d can be regarded as the limit of ∗ − , ⊛ q and d q as ~ → q → ζ ( d ( φ ( w ∗ w )) ⊛ z n ) + X k + l = nk,l ≥ ζ ( d ( φ ( w )) ⊛ z k ) ζ ( d ( φ ( w )) ⊛ z l ) = 0 ( w , w ∈ h > ) . (4.1)Setting n = 1 we obtain linear relations for MZV’s ζ ( z ◦ d ( φ ( w ∗ w ))) = 0 ( w , w ∈ h > ) . (4.2)4.2. A q -analogue of Kawashima’s relation.Proposition 4.1. For w , w ∈ h > and n ≥ , the following relation holds: ζ q ( d q ( φ ( w ∗ − w )) ⊛ q z n ) + X k + l = nk,l ≥ ζ q ( d q ( φ ( w )) ⊛ q z k ) ζ q ( d q ( φ ( w )) ⊛ q z l ) = 0 . (4.3) Proof. If w ∈ h > , we have ∇ q ( S w )(0) = 0 and v ( ∇ q ( S w )( n )) = v ( − s φ ( w ) ( n − ≥ n ( n > . from Proposition 2.1. Hence ∇ q ( S w ) satisfies the condition (3.5), and the series F w ( z ) := f ∇ ( S w ) ( z ) is well-defined. Expanding F w ( z ) at z = 1 using Proposition 3.3and Proposition 2.7, we see that F w ( z ) = ∞ X m =1 ∞ X n =1 ∇ ( S w )( n ) a z m ( n − ! (cid:18) z − − q (cid:19) m (4.4) = − ∞ X m =1 ∞ X n =1 s φ ( w ) ( n − a z m ( n − ! (cid:18) z − − q (cid:19) m = − ∞ X m =1 ∞ X n =0 a d q ( φ ( w )) ⊛ q z m ( n ) ! (cid:18) z − − q (cid:19) m = − ∞ X m =1 ζ q ( d q ( φ ( w )) ⊛ q z m ) (cid:18) z − − q (cid:19) m . From Lemma 2.3 and Proposition 3.4, we have F w F w = F w ∗ − w for w , w ∈ h > .Substituting (4.4), we get the relation (4.3). (cid:3) Corollary 4.2.
For w , w ∈ h > , we have ζ ⋆q ( z ◦ φ ( w ∗ − w )) = 0 . Proof.
Set n = 1 in (4.3). Then it follows from Lemma 2.4 and Corollary 2.6. (cid:3) Let us rewrite the quadratic relation (4.3) into a similar form to Kawashima’srelation (4.1). Consider the mapΨ := φ d − q d φ : h → h . Note that Ψ(1) = 1.
Lemma 4.3.
Let w ∈ h . Then Ψ( z w ) = z Ψ( w ) , Ψ( z i w ) = ( z − ~ z ) ◦ Ψ( z i − w ) ( i ≥ . Proof.
Note that φ ( z i w ) = z i − ( z ◦ φ ( w )) for i ≥ w ∈ h . Since the maps d q and d commute with the action ◦ of z , we get Ψ( z w ) = z Ψ( w ). To show thesecond formula, use the identity( d − q d )( z w ) = ( z − ~ z ) ◦ ( d − q d )( w ) ( w ∈ h )following from the definition of d q and d . (cid:3) Proposition 4.4.
Let w ∈ h . Then Ψ( z i w ) = ξ i Ψ( w ) for i ≥ , where ξ i ∈ z isgiven by ξ i := i − X k =0 (cid:18) i − k (cid:19) ( − ~ ) i − − k z k +1 . Proof.
Proposition 4.3 implies that it suffices to prove ( z − ~ z ) ◦ ξ i = ξ i +1 for i ≥ (cid:3) Proposition 4.5.
Let w , w ∈ h . Then Ψ( w ) ∗ − Ψ( w ) = Ψ( w ∗ w ) . Proof.
We can assume that w and w are words. Let us prove the proposition byinduction on the depth of w and w . If w = 1 or w = 1, it is trivial. Set w = z i w ′ and w = z j w ′ . From Proposition 4.4 and the induction hypothesis, we findΨ( w ) ∗ − Ψ( w ) = ( ξ i Ψ( w ′ )) ∗ − ( ξ j Ψ( w ′ ))= ξ i Ψ( w ′ ∗ w ) + ξ j Ψ( w ∗ w ′ ) + ( ξ i ◦ − ξ j )Ψ( w ′ ∗ w ′ ) . UADRATIC RELATIONS FOR q MZV 13
By direct calculation we see that ξ i ◦ − ξ j = − ξ i + j for i, j ≥
1. Therefore we getΨ( w ) ∗ − Ψ( w ) = ξ i Ψ( w ′ ∗ w ) + ξ j Ψ( w ∗ w ′ ) − ξ i + j Ψ( w ′ ∗ w ′ )= Ψ( z i ( w ′ ∗ w ) + z j ( w ∗ w ′ ) − z i + j ( w ′ ∗ w ′ ))= Ψ( w ∗ w ) . (cid:3) Now we are ready to derive a q -analogue of Kawashima’s relations: Theorem 4.6.
For w , w ∈ h > and n ≥ , the following relation holds: ζ q ( d ( φ ( w ∗ w )) ⊛ q z n ) + X k + l = nk,l ≥ ζ q ( d ( φ ( w )) ⊛ q z k ) ζ q ( d ( φ ( w )) ⊛ q z l ) = 0 . (4.5) Proof.
Substitute Ψ( w i ) into w i ( i = 1 ,
2) in the quadratic relation (4.3). Then weget the relation (4.5) using Proposition 4.5 and d q φ Ψ = dφ . (cid:3) Setting n = 1 in (4.5), we obtain linear relations for q MZV’s in the same form as(4.2):
Corollary 4.7.
For w , w ∈ h > , we have ζ q ( z ◦ d ( φ ( w ∗ w ))) = 0 . Acknowledgments
The research of the author is supported by Grant-in-Aid for Young Scientists (B)No. 20740088. The author is grateful to Yasuo Ohno, Jun-ichi Okuda and TatsushiTanaka for helpful informations.
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