Special values of finite multiple harmonic q-series at roots of unity
aa r X i v : . [ m a t h . N T ] J u l Special values of finite multiple harmonic q -seriesat roots of unity Henrik Bachmann ∗ , Yoshihiro Takeyama † , Koji Tasaka ‡ Abstract
We study special values of finite multiple harmonic q -series at roots of unity.These objects were recently introduced by the authors and it was shown that theyhave connections to finite and symmetric multiple zeta values and the Kaneko-Zagier conjecture. In this note we give new explicit evaluations for finite multipleharmonic q -series at roots of unity and prove Ohno-Zagier-type relations for them. Finite multiple harmonic q -series at roots of unity were introduced by the authors in[1]. The motivation to study these object is their connection to finite and symmetricmultiple zeta values and a reinterpretation of the conjecture of Kaneko and Zagier in[4]. In this paper we concentrate on the finite multiple harmonic q -series at roots ofunity themselves and give explicit evaluations and Ohno-Zagier-type relations for them.For an index k = ( k , . . . , k r ) ∈ ( Z ≥ ) r , a natural number n ≥ q satisfying q m = 1 for n > m >
0, the finite multiple harmonic q -series andtheir star-versions are defined by z n ( k ; q ) = z n ( k , . . . , k r ; q ) = X n>m > ··· >m r > q ( k − m . . . q ( k r − m r [ m ] k q . . . [ m r ] k r q ,z ⋆n ( k ; q ) = z ⋆n ( k , . . . , k r ; q ) = X n>m ≥···≥ m r > q ( k − m . . . q ( k r − m r [ m ] k q . . . [ m r ] k r q , ∗ email : [email protected], Nagoya University † email : [email protected], University of Tsukuba ‡ email : [email protected], Aichi Prefectural University m ] q is the q -integer [ m ] q = − q m − q . For a primitive n -th root of unity ζ n the values z n ( k ; ζ n ) and z ⋆n ( k ; ζ n ) are elements in the cyclotomic field Q ( ζ n ). The first result ofthis paper are the following evaluations of z n ( k, . . . , k ; ζ n ). Theorem 1.1.
For all k, r ≥ and any n -th primitive root of unity ζ n we have z n ( { k } r ; ζ n ) ∈ (1 − ζ n ) kr Q and in particular z n ( { } r ; ζ n ) = 1 n (cid:18) nr + 1 (cid:19) (1 − ζ n ) r ,z n ( { } r ; ζ n ) = ( − r n · ( r + 1) (cid:18) n + r r + 1 (cid:19) (1 − ζ n ) r ,z n ( { } r ; ζ n ) = 1 n ( r + 1) (cid:18)(cid:18) n + 2 r + 13 r + 2 (cid:19) + ( − r (cid:18) n + r r + 2 (cid:19)(cid:19) (1 − ζ n ) r . The second result of this paper are Ohno-Zagier-type relations for z n ( k ; ζ n ) and z ⋆n ( k ; ζ n ). For an index k = ( k , . . . , k r ) denote by wt( k ) = k + · · · + k r the weight and by dep( k ) = r the depth . In [5], Ohno and Zagier introduced the height byht( k ) = { a ∈ { , , . . . , r } | k a ≥ } . They proved an explicit formula for the gen-erating function of the sum of multiple zeta values of fixed weight, depth and height.To state the Ohno-Zagier-type relations for finite multiple harmonic q -series at rootsof unity we first define their modified versions by z n ( k ; ζ n ) := (1 − ζ n ) − wt( k ) z n ( k ; ζ n ) , z ⋆n ( k ; ζ n ) := (1 − ζ n ) − wt( k ) z ⋆n ( k ; ζ n ) . For positive integers k, r, s ≥
0, let I ( k, r, s ) be the set of indices of weight k , depth r and height s . We denote the generating function of the sum of the modified versions offixed weight, depth and height by F n ( x, y, z ) = 1 + ∞ X r =1 r X s =0 ∞ X k = r + s X k ∈ I ( k,r,s ) z n ( k ; ζ n ) x k − r − s y r − s z s ,F ⋆n ( x, y, z ) = 1 + ∞ X r =1 r X s =0 ∞ X k = r + s X k ∈ I ( k,r,s ) z ⋆n ( k ; ζ n ) x k − r − s y r − s z s . (1.1) Theorem 1.2.
The sum of z n ( k ) (or z ⋆n ( k ) ) of fixed weight, depth and height is arational number. More explicitly we have F n ( x, y, z ) = U n ( x, y, z ) , F ⋆n ( x, y, z ) = U n ( x, − y, − z ) − , here U n ( x, y, z ) = x (1 + x ) n − × X a,b ≥ a + b ≤ n − n − a − b (cid:18) n − a − b (cid:19)(cid:18) n − b − a (cid:19) (1 + x ) a (1 + y ) b ( xy − z ) n − − a − b . (1.2)In [1, Theorem 1.2] it was shown that for any index k ∈ ( Z ≥ ) r , the limit ξ ( k ) = lim n →∞ z n ( k ; e πi/n )exists in C . The real part of ξ ( k ) has a connection to symmetric multiple zeta values. Asan application of Theorems 1.1 and 1.2 we will give explicit evaluations of ξ ( k, . . . , k )for k = 1 , , ξ ( k ). Corollary 1.3.
For all k ≥ r ≥ we have X k ∈ I ( k,r ) ξ ( k ) = − ( − πi ) k ( k + 1)! r X j =1 (cid:18) k + 1 j (cid:19) B k +1 − j , where I ( k, r ) denotes the set of indices of weight k and depth r and where B k is the k -th Bernoulli number with the convention B = − . The contents of this paper are as follows. In Section 2 we consider the values z n ( k, . . . , k ; ζ n ) and give the proof of Theorem 1.1. The Ohno-Zagier-type relations andthe proof of Theorem 1.2 will be given in Section 3. In Section 4 we present evaluationsand the Ohno-Zagier-type relations for the values ξ ( k ) and give the proof of Corollary1.3. Acknowledgments
This work was partially supported by JSPS KAKENHI GrantNumbers 16F16021, 16H07115, 18K13393 and 18K03233. z n ( k, . . . , k ; ζ n ) For depth one it was shown in [1] (see also Corollary 3.4), that for all k ≥ z n ( k ; ζ n ) = − β k ( n − ) n k k ! (1 − ζ n ) k , (2.1)3here β k ( x ) ∈ Q [ x ] is the degenerate Bernoulli number defined by Carlitz in [3]. Since β k ( n − ) n k ∈ Q [ n ], the z n ( k ; ζ n ) are polynomials in n . For example, we have z n (1; ζ n ) = n −
12 (1 − ζ n ) , z n (2; ζ n ) = − n −
112 (1 − ζ n ) ,z n (3; ζ n ) = n −
124 (1 − ζ n ) , z n (4; ζ n ) = ( n − n − − ζ n ) . The formula (2.1) can be seen as an analogue of the formula by Euler for the Riemannzeta values ζ ( k ) = − B k k ! ( − πi ) k for even k . The multiple zeta values defined for k ≥ , k , . . . , k r ≥ ζ ( k , . . . , k r ) = X m > ··· >m r > m k · · · m k r r . also have explicit evaluations for even k and k = · · · = k r = k . Theorem 1.1 can beseen as an analogue of these formulas. We have for example ζ ( { } r ) = π r (2 r + 1)! , which is an easy consequence of the product formula of the sine ∞ X j =0 ζ ( { } j ) T j +1 = T ∞ Y m =1 (cid:18) T m (cid:19) = sin ( πiT ) πi = ∞ X j =0 π j (2 j + 1)! T j +1 . We will use a similar idea for the proof of Theorem 1.1.
Proof of Theorem 1.1.
We will prove explicit formulas for the modified versions z ( { k } r ; ζ n )and then obtain the result in Theorem 1.1 by multiplying with (1 − ζ n ) kr . By definitionof the values z n ( { k } r ; ζ n ) we derive n − X r =0 z n ( { k } r ; ζ n ) X r = n − Y j =1 ζ ( k − jn (1 − ζ jn ) k X ! = 1 n k n − Y j =1 (cid:0) (1 − ζ jn ) k + ζ ( k − jn X (cid:1) , where we used Q n − j =1 (1 − ζ jn ) = n at the last equation. For i = 1 , . . . , k we denote by α i the roots of the polynomial (1 − Y ) k + Y ( k − X , i.e. k Y i =1 ( α i − Y ) := (1 − Y ) k + Y ( k − X . α i can not be computed explicitly but by definition they satisfye j ( α , . . . , α k ) = (cid:18) kj (cid:19) + ( − k − δ j, X , (2.2)with the elementary symmetric polynomials e j ( x , . . . , x k ) defined by k X j =0 ( − j e j ( x , . . . , x k ) T k − j := k Y j =1 ( T − x j ) . Using Q n − j =1 ( α i − ζ jn ) = α ni − α i − and Q ki =1 ( α i −
1) = X we obtain1 n k n − Y j =1 k Y i =1 ( α i − ζ jn ) = 1 n k k Y i =1 α ni − α i − n k X k Y i =1 ( α ni − . Now consider also the generating series of this over all n ∞ X n =1 n k − Y n n − X r =0 z ( { k } r ; ζ n ) X r = 1 X ∞ X n =1 Y n n k Y i =1 ( α ni − X ∞ X n =1 Y n n ( − k + k X l =1 ( − k − l X ≤ i < ···
1. For the cases k = 1 , , F , ( X, Y ) = F , ( X, Y ) = 1 − Y ,F , ( X, Y ) = F , ( X, Y ) = 1 − Y , F , ( X, Y ) = (1 − Y ) + XY ,F , ( X, Y ) = F , ( X, Y ) = 1 − Y , F , ( X, Y ) = (1 − Y ) − XY ,F , ( X, Y ) = (1 − Y ) + XY . Plugging these into (2.3), we get for example for the k = 3 case1 X log Y l =0 F ,l ( X, Y ) ( − l ! = 1 X log (cid:18) (1 − Y ) + XY (1 − Y ) − XY (cid:19) = 1 X (cid:18) log (cid:18) XY (1 − Y ) (cid:19) − log (cid:18) − XY (1 − Y ) (cid:19)(cid:19) = 1 X − ∞ X m =1 ( − m − m (cid:18) XY (1 − Y ) (cid:19) m + ∞ X m =1 n (cid:18) XY (1 − Y ) (cid:19) m ! Using for l ≥ − Y ) l = ∞ X j =0 (cid:18) l − jl − (cid:19) Y j , we obtain by (2.3) the formula for z n ( { } r ; ζ n ) as stated in Theorem 1.1. The formulasfor z n ( { } r ; ζ n ) and z n ( { } r ; ζ n ) follow similarly.Notice that above proof works for every k , i.e. it is possible to obtain explicitformulas for z n ( { k } r ; ζ n ). But these formulas might get quite complicated for larger k . Remark . Additionally to the above formulas we have the following observations.For all a, b ≥ z n ( { } a , , { } b ; ζ n ) + z n ( { } b , , { } a ; ζ n ) ? = − n (cid:18) n + 1 a + b + 3 (cid:19) (1 − ζ n ) a + b +2 ,z n ( { } a , , { } b ; ζ n ) + z n ( { } b , , { } a ; ζ n ) ? = − ( − a + b ( a + b + 2) n (cid:18) n + a + b + 12( a + b ) + 3 (cid:19) (1 − ζ n ) a + b )+3 , but so far the authors were not able to prove these formulas.6 Ohno-Zagier-type relation
The proof of Theorem 1.2 is essentially the same as that of Ohno-Zagier’s relation formultiple zeta values [5] (see also [6, 7]) except for the following point. In both proofs themain ingredient is the generating function of type (1.1). While Ohno-Zagier’s relationfollows from its explicit formula in terms of the hypergeometric function and its specialvalue, our formula follows from a consistency condition for the q -difference equation ofwhich the generating function gives a unique polynomial solution. q -analogue of finite multiple polylogarithm of one variable Throughout this subsection, q denotes a complex number. For a positive integer n andan index k = ( k , . . . , k r ), we define the function L ( n ) k ( t ; q ) and its star version L ⋆, ( n ) k ( t ; q )by L ( n ) k ( t ; q ) = X n>m > ··· >m r > t m Q ri =1 (1 − q m i ) k i ,L ⋆, ( n ) k ( t ; q ) = X n>m ≥···≥ m r > t m Q ri =1 (1 − q m i ) k i . Note that when q ∈ C with | q | < q = ζ n a n -th primitive root of unity, L ( n ) k ( t ; q ) and L ⋆, ( n ) k ( t ; q ) are polynomials in t whose degree is less than n , and they satisfy L ( n ) k ( t ; q ) = O ( t dep( k ) ) and L ⋆, ( n ) k ( t ; q ) = O ( t ) as t →
0. Taking the limit as n → ∞ we recover the q -multiple polylogarithms defined in [2, 8].Hereafter we fix a positive integer n and omit the superscript ( n ) in L ( n ) k ( t ; q ) and L ⋆, ( n ) k ( t ; q ).We denote by I ( k, r, s ) the set of admissible indices of weight k , depth r and height s . For non-negative integers k, r and s , set G ( k, r, s ; t ; q ) = X k ∈ I ( k,r,s ) L k ( t ; q ) , G ( k, r, s ; t ; q ) = X k ∈ I ( k,r,s ) L k ( t ; q ) ,G ⋆ ( k, r, s ; t ; q ) = X k ∈ I ( k,r,s ) L ⋆ k ( t ; q ) , G ⋆ ( k, r, s ; t ; q ) = X k ∈ I ( k,r,s ) L ⋆ k ( t ; q ) , where I ( k, r, s ) is the set of indices of weight k , depth r and height s . If I ( k, r, s )(resp. I ( k, r, s )) is empty, G ( k, r, s ; t ; q ) and G ⋆ ( k, r, s ; t ; q ) (resp. G ( k, r, s ; t ; q ) and G ⋆ ( k, r, s ; t ; q )) are zero, while we set by definition G (0 , , t ; q ) = G ⋆ (0 , , t ; q ) = 1and G (0 , , t ; q ) = G ⋆ (0 , , t ; q ) = 0. 7onsider the generating functionsΦ( t ; q ) = Φ( u, v, w ; t ; q ) = X k,r,s ≥ G ( k, r, s ; t ; q ) u k − r − s v r − s w s , Φ ( t ; q ) = Φ ( u, v, w ; t ; q ) = X k,r,s ≥ G ( k, r, s ; t ; q ) u k − r − s v r − s w s − , Φ ⋆ ( t ; q ) = Φ ⋆ ( u, v, w ; t ; q ) = X k,r,s ≥ G ⋆ ( k, r, s ; t ; q ) u k − r − s v r − s w s , Φ ⋆ ( t ; q ) = Φ ⋆ ( u, v, w ; t ; q ) = X k,r,s ≥ G ⋆ ( k, r, s ; t ; q ) u k − r − s v r − s w s − . Note that they are polynomials in t whose coefficients are formal power series in u, v and w , and whose degree is less than n . Moreover, from the definition, it holds thatΦ (0; q ) = Φ ⋆ (0; q ) = 0.The coefficients of the generating functions Φ(1; q ) and Φ ⋆ (1; q ) can be written interms of z n ( k ; q ) and z ⋆n ( k ; q ). Lemma 3.1.
Set u = x x , v = y − z x , w = z (1 + x ) . (3.1) Then we have Φ( u, v, w ; 1; q ) = 1 + ∞ X r =1 r X s =0 ∞ X k = r + s X k ∈ I ( k,r,s ) z n ( k ; q ) x k − r − s y r − s z s (3.2) and Φ ⋆ ( u, v, w ; 1; q ) = 1 + ∞ X r =1 r X s =0 ∞ X k = r + s X k ∈ I ( k,r,s ) z ⋆n ( k ; q ) x k − r − s y r − s z s . (3.3) Proof.
Since we have1(1 − q ) k = k X a =1 (cid:18) k − a − (cid:19) q ( a − m (1 − q m ) a ( k ≥ , m ≥ , it holds that L k ,...,k r (1; q ) = k X a =1 · · · k r X a r =1 ( r Y j =1 (cid:18) k j − a j − (cid:19)) z n ( a , . . . , a r ; q ) . (3.4)8sing the above formula we find thatΦ( u, v, w ; 1; q )= n − X r =1 (cid:16) vu (cid:17) r X a ,...,a r ≥ z n ( a , . . . , a r ; q ) r Y j =1 ∞ X k = a j (cid:18) k − a j − (cid:19) u k (cid:16) wuv (cid:17) θ ( k ≥ , where θ (P) = 1 if P is true and θ (P) = 0 if P is false. It holds that ∞ X k = a (cid:18) k − a − (cid:19) u k (cid:16) wuv (cid:17) θ ( k ≥ = (cid:18) u − u (cid:19) a (cid:16) − u + wv (cid:17) (cid:18) wu ( v + w − uv ) (cid:19) θ ( a ≥ ( a ≥ . Thus we getΦ( u, v, w ; 1; q )= X a dep( a ) ≤ n − z n ( a ; q ) (cid:18) u − u (cid:19) wt( a ) n vu (cid:16) − u + wv (cid:17)o dep( a ) (cid:18) wu ( v + w − uv ) (cid:19) ht( a ) . Changing the variables u, v and w to x, y and z by (3.1), we get the desired formula(3.2).For the star version, the formula (3.4) still holds if L k ,...,k r (1; q ) and z n ( a , . . . , a r ; q )are replaced by L ⋆k ,...,k r (1; q ) and z ⋆n ( a , . . . , a r ; q ), respectively. Then one can check theequality (3.3) in the same way.We define the q -difference operator D q by( D q f )( t ) = 1 t ( f ( t ) − f ( qt )) . The above generating functions satisfy the following q -difference equations. Lemma 3.2.
We have qt (1 − t ) D q Φ ( t ; q ) + { (1 − q − u )(1 − t ) − vt } D q Φ ( t ; q ) + ( uv − w )Φ ( t ; q )= 1 − Φ(1; q ) t n − , (3.5) qt (1 − t ) D q Φ ⋆ ( t ; q ) + t { (1 − q − u )(1 − t ) − v } D q Φ ⋆ ( t ; q ) + ( uv − w )Φ ⋆ ( t ; q )= t − Φ ⋆ (1; q ) t n . roof. We make use of the following formulas: D q L k ,...,k r ( t ) = t L k − ,k ,...,k r ( t ) ( k ≥ − t (cid:0) L k ,...,k r ( t ) − t n − L k ,...k r (1) (cid:1) ( k = 1)and D q L ⋆k ,...,k r ( t ) = t L ⋆k − ,k ,...,k r ( t ) ( k ≥ t (1 − t ) (cid:0) L ⋆k ,...,k r ( t ) − t n − L ⋆k ,...,k r (1) (cid:1) ( k = 1 , r ≥ − t n − − t ( k = 1 , r = 1) . Since the proof is similar, we only prove (3.5). From the above formulas, we see that D q G ( k, r, s ; t ; q )= 1 t ( G ( k − , r, s ; t ; q ) + G ( k − , r, s − t ; q ) − G ( k − , r, s − t ; q )) , D q ( G ( k, r, s ; t ; q ) − G ( k, r, s ; t ; q ))= 11 − t (cid:0) G ( k − , r − , s ; t ; q ) − t n − G ( k − , r − , s ; 1; q ) (cid:1) , and hence D q Φ ( t ; q ) = 1 t (cid:26) ( u − wv )Φ ( t ; q ) + 1 v (Φ( t ; q ) − (cid:27) , D q Φ( t ; q ) − w D q Φ ( t ; q ) = v − t (cid:0) Φ( t ; q ) − t n − Φ(1; q ) (cid:1) . Eliminate Φ( t ; q ) using the Leibniz rule D q ( f ( t ) g ( t )) = D q ( f ( t )) g ( t ) + f ( qt ) D q ( g ( t )) . Then we get (3.5).
Proposition 3.3.
We have
Φ(1; q ) = n − Y j =1 P ( q j )(1 − q j )(1 − u − q j ) , (3.6)10 nd Φ ⋆ (1; q ) = n − Y j =1 (1 − q j )(1 − u − q j ) P ⋆ ( q j ) , (3.7) where P ( X ) and P ⋆ ( X ) are the quadratic polynomials given by P ( X ) = (1 − u − X )(1 + v − X ) + w, P ⋆ ( X ) = (1 − u − X )(1 − v − X ) − w. (3.8) Proof.
Set Φ ( t ; q ) = P n − j =1 c j t j and substitute it into the equation (3.5). Then we findthat (1 − q )(1 − q − u ) c = 1 , (3.9)(1 − q j +1 )(1 − u − q j +1 ) c j +1 = P ( q j ) c j (1 ≤ j ≤ n − , (3.10) P ( q n − ) c n − = Φ(1; q ) . (3.11)From the initial condition (3.9) and the recurrence relation (3.10), we get c n − = Q n − j =1 P ( q j ) Q n − j =1 (1 − q j )(1 − u − q j ) . Hence the consistency between the above formula and (3.11) implies (3.6). The verifi-cation of (3.7) is similar.
Now we set q = ζ n . By Lemma 3.1, we know thatΦ( u, v, w ; 1; ζ n ) = F n ( x, y, z ) , Φ ⋆ ( u, v, w ; 1; ζ n ) = F ⋆n ( x, y, z ) , where u, v, w are given by (3.1). Therefore Theorem 1.2 follows from the equalitiesΦ(1; ζ n ) = U n ( x, y, z ) and Φ ⋆ (1; ζ n ) = U n ( x, − y, − z ) − . Let us rewrite the right hand side of (3.6) with q = ζ n . After the change of variables(3.1), the polynomial P ( X ) defined by (3.8) becomes P ( X ) = 1 + y x − (cid:18) y + 1 − z x (cid:19) X + X . α and β such that α + β = (1 + x )(1 + y ) + 1 − z, αβ = (1 + x )(1 + y ) . Then it holds that P ( X ) = (cid:18) α x − X (cid:19) (cid:18) β x − X (cid:19) . Using (3.1) and the equality Q n − j =1 ( T − ζ jn ) = ( T n − / ( T − q = ζ n as n − Y j =1 P ( ζ jn )(1 − ζ jn )(1 − u − ζ jn ) = x (1 + x ) n − xy − z α n + β n − (1 + x ) n − (1 + y ) n n . Thus we find thatΦ(1; ζ n ) = x (1 + x ) n − xy − z α n + β n − (1 + x ) n − (1 + y ) n n . (3.12)To calculate the last factor we consider the generating function ∞ X n =1 α n + β n − (1 + x ) n − (1 + y ) n n T n . It is equal tolog (cid:18) (1 − (1 + x ) T )(1 − (1 + y ) T )(1 − αT )(1 − βT ) (cid:19) = − log (cid:18) − ( xy − z ) T (1 − (1 + x ) T )(1 − (1 + y ) T ) (cid:19) = ∞ X n =1 n (cid:18) ( xy − z ) T (1 − (1 + x ) T )(1 − (1 + y ) T ) (cid:19) n . Expand it into a power series of T . Then the coefficient of T n is given by X a,b ≥ a + b ≤ n − n − a − b (cid:18) n − a − b (cid:19)(cid:18) n − b − a (cid:19) (1 + x ) a (1 + y ) b ( xy − z ) n − a − b . Thus we get Φ(1; ζ n ) = U n ( x, y, z ).The proof for the remaining equality Φ ⋆ (1; ζ n ) = U n ( x, − y, − z ) − is similar. Thepolynomial P ⋆ ( X ) is obtained from P ( X ) by the change of variables ( v, w ) → ( − v, − w ).It corresponds to the change ( y, z ) → ( − y, − z ) under the transform (3.1). Since12(1; ζ n ) = U n ( x, y, z ), we get Φ ⋆ (1; ζ n ) = U n ( x, − y, − z ) − . This completes the proof ofTheorem 1.2.We end this subsection by considering a specialization of Theorem 1.2. Corollary 3.4. (i) It holds that ∞ X k =1 z n ( k ; ζ n ) x k = nx − (1 + x ) n + 1 . (3.13) (ii) Suppose that k ≥ r and n > r > , and denote by I ( k, r ) the set of indices of weight k and depth r . Then we have X k ∈ I ( k,r ) z n ( k ; ζ n ) = r X j =1 n (cid:18) nj (cid:19) z n ( k + 1 − j ; ζ n ) . (3.14) Proof.
Setting z = xy we have F n ( x, y, xy ) = 1 + n − X r =1 ∞ X k = r X k ∈ I ( k,r ) z n ( k ; ζ n ) x k − r y r . We rewrite U n ( x, y, xy ) as follows. Since it holds that X a,b ≥ a + b = n − (cid:18) n − a − b (cid:19)(cid:18) n − b − a (cid:19) (1 + x ) a (1 + y ) b = X a,b ≥ a + b = n − (1 + x ) a (1 + y ) b = (1 + x ) n − x + n − X r =1 y r x − r (1 + x ) n − x − r X j =1 (cid:18) nj (cid:19) x j − ! , we have U n ( x, y, xy ) = 1 + n − X r =1 y r x − r − x (1 + x ) n − r X j =1 (cid:18) nj (cid:19) x j − ! . Comparing the coefficient of y r in the equality F n ( x, y, xy ) = U n ( x, y, xy ), we see that ∞ X k = r X k ∈ I ( k,r ) z n ( k ; ζ n ) x k = 1 − x (1 + x ) n − r X j =1 (cid:18) nj (cid:19) x j − (3.15)13or r ≥
1. Setting r = 1 we obtain (3.13).From (3.13) we see that x (1 + x ) n − n − ∞ X l =1 z n ( l ; ζ n ) x l ! . Substituting it into (3.15) we obtain the equality ∞ X k = r X k ∈ I ( k,r ) z n ( k ; ζ n ) x k = 1 − − ∞ X l =1 z n ( l ; ζ n ) x l ! r − X j =1 n (cid:18) nj + 1 (cid:19) x j ! . It implies the desired equality (3.14).
In this section we give applications to the numbers ξ ( k ) and ξ ⋆ ( k ), which were definedin [1, Theorem 1.2] by the limits ξ ( k ) = lim n →∞ z n ( k ; e πi/n ) , ξ ⋆ ( k ) = lim n →∞ z ⋆n ( k ; e πi/n ) . The real part of ξ ( k ) is congruent to the symmetric multiple zeta values modulo ζ (2).From (2.1) it follows that for all k ≥ ξ ( k ) = − B k k ! ( − πi ) k , (4.1)where B k is the k -th Bernoulli number with the convention B = − . As a consequenceof Section 2 we obtain the following. Corollary 4.1.
For all k, r ≥ we have ξ ( { k } r ) ∈ ( − πi ) kr Q and in particular ξ ( { } r ) = ( − πi ) r r + 1 ,ξ ( { } r ) = 2 r π r ( r + 1)(2 r + 1)! = 2 r r + 1 ζ ( { } r ) ,ξ ( { } r ) = (1 + ( − r )( − πi ) r ( r + 1)(3 r + 2)! . Proof.
This follows directly from Theorem 1.1 together with the fact that n (1 − e πi/n )goes to − πi as n → ∞ . 14s a consequence of the results in Section 3 we obtain the following Ohno-Zagier-type relations for the values ξ ( k ). Proposition 4.2.
For positive integers k, r and s , the sum of ξ ( k ) (or ξ ⋆ ( k ) ) over I ( k, r, s ) belongs to ( − πi ) k Q . More explicitly we have the following formula for thegenerating functions. Define e F ( x, y, z ) = 1 + ∞ X r =1 r X s =0 ∞ X k = r + s ( − πi ) − k X k ∈ I ( k,r,s ) ξ ( k ) x k − r − s y r − s z s , e F ⋆ ( x, y, z ) = 1 + ∞ X r =1 r X s =0 ∞ X k = r + s ( − πi ) − k X k ∈ I ( k,r,s ) ξ ⋆ ( k ) x k − r − s y r − s z s . Then we have e F ( x, y, z ) = e U ( x, y, z ) , e F ⋆ ( x, y, z ) = e U ( x, − y, − z ) − , where e U ( x, y, z ) = e y/ x sinh ( x/
2) cosh ( p ( x + y ) − z ) − cosh ( ( x − y )) xy − z . Proof.
Let F n ( x, y, z ) be the generating function defined by (1.1). The coefficient of x k − r − s y r − s z s in F n ( x/n, y/n, z/n ) is equal to n − k P k ∈ I ( k,r,s ) z n ( k ; e πi/n ). From thedefinition of ξ ( k ), it holds thatlim n →∞ n − wt( k ) z n ( k ; e πi/n ) = ( − πi ) − wt( k ) ξ ( k ) . Hence we see that( − πi ) − k X k ∈ I ( k,r,s ) ξ ( k )= lim n →∞ I x − ( k − r − s ) − dx πi I y − ( r − s ) − dy πi I z − s − dz πi U n ( xn , yn , zn ) , (4.2)where U n is defined by (1.2) and H denotes the integration around the origin.To justify the interchange of limit and integration we estimate U n ( x/n, y/n, z/n ).There exists a positive constant c such that0 < e t − (1 + t ) t ≤ ct (0 < t ≤ .
15e choose a positive constant ǫ so that c ǫ < /
2, and the integration contours to be | x | = ǫ, | y | = ǫ and | z | = ǫ . We decompose into two parts U n ( x/n, y/n, z/n ) = J J ,where J = x (1 + x/n ) n − ,J = 1 n X a,b ≥ a + b ≤ n − n − a − b (cid:18) n − a − b (cid:19)(cid:18) n − b − a (cid:19) (cid:16) xn (cid:17) a (cid:16) yn (cid:17) b (cid:18) xy − zn (cid:19) n − − a − b . First we see that, on the circle | x | = ǫ , (cid:12)(cid:12) J − − (cid:12)(cid:12) ≤ ǫ n(cid:16) ǫn (cid:17) n − (1 + ǫ ) o ≤ ǫ ( e ǫ − − ǫ ) ≤ cǫ < . Hence | J | <
2. Second J is estimated by using | x | = | y | = ǫ and | z | = ǫ as | J | ≤ n X a,b ≥ a + b ≤ n − n − a − b (cid:18) n − a − b (cid:19)(cid:18) n − b − a (cid:19) (cid:16) ǫn (cid:17) a + b (cid:18) ǫ n (cid:19) n − − a − b ≤ n n − X m =0 (cid:16) ǫn (cid:17) m (cid:18) ǫ n (cid:19) n − m − X a,b ≥ a + b = m (cid:18) n − a − b (cid:19)(cid:18) n − b − a (cid:19) ≤ e ǫ n − X m =0 n n − m (cid:18) n − m − m (cid:19) (cid:18) ǫ n (cid:19) n − m − . Here we used X a,b ≥ a + b = m (cid:18) n − a − b (cid:19)(cid:18) n − b − a (cid:19) = (cid:18) n − m − m (cid:19) ( n > m ≥ , to obtain the last inequality. Since1 n n − m (cid:18) n − m − m (cid:19) = (cid:18) n − m (cid:19) ( n − m − n − m − n − m − Y a =1 (cid:18) − m + an (cid:19) ≤ (cid:18) n − m (cid:19) n − m − ( n − m − n − m − ≤ (cid:18) n − m (cid:19) ,
16e find that | J | ≤ e ǫ n − X m =0 (cid:18) n − m (cid:19) (cid:18) ǫ n (cid:19) n − m − = e ǫ (cid:18) ǫ n (cid:19) n − ≤ e ǫ +2 ǫ . Thus we see that (cid:12)(cid:12)(cid:12) U n ( xn , yn , zn ) (cid:12)(cid:12)(cid:12) ≤ e ǫ +2 ǫ on | x | = | y | = ǫ, | z | = ǫ . Therefore we can interchange the limit and the integration of(4.2). As a consequence we find that e F ( x, y, z ) = lim n →∞ U n (cid:16) xn , yn , zn (cid:17) . To calculate the limit in the right hand side, we make use of the expression (3.12).It holds that U n (cid:16) xn , yn , zn (cid:17) = 1 xy − z x (1 + x/n ) n − n ˜ α n + ˜ β n − (cid:16) xn (cid:17) n − (cid:16) yn (cid:17) n o , where ˜ α and ˜ β are determined from˜ α + ˜ β = (cid:16) xn (cid:17) (cid:16) yn (cid:17) + 1 − zn , ˜ α ˜ β = (cid:16) xn (cid:17) (cid:16) yn (cid:17) . Then we see that the asymptotic behavior of ˜ α and ˜ β are given by1 + 12 n (cid:16) x + y ± p ( x + y ) − z (cid:17) + o ( 1 n ) ( n → ∞ ) . Hence we find thatlim n →∞ U n (cid:16) xn , yn , zn (cid:17) = 1 xy − z xe x − (cid:26) e ( x + y ) / cosh ( 12 p ( x + y ) − z ) − e x − e y (cid:27) = 1 xy − z xe y/ sinh ( x/ (cid:26) cosh ( 12 p ( x + y ) − z ) − cosh ( 12 ( x − y )) (cid:27) . Thus we get the equality e F ( x, y, z ) = e U ( x, y, z ).By the same argument as before we see that e F ⋆ ( x, y, z ) − = lim n →∞ U n (cid:16) xn , − yn , − zn (cid:17) . Hence it also holds that e F ⋆ ( x, y, z ) = e U ( x, − y, − z ) − .17e end this note by giving the proof of the sum-formula for the ξ ( k ). Proof of Corollary 1.3.
Using Stirling’s formula, we see thatlim n →∞ (1 − e πi/n ) j − n (cid:18) nj (cid:19) = ( − πi ) j − j ! . Combining this with Corollary 3.4, we obtain X k ∈ I ( k,r ) ξ ( k ) = r X j =1 ( − πi ) j − j ! ξ ( k + 1 − j ) . Using (4.1) for the evaluation of ξ ( k + 1 − j ) we obtain the formula in Corollary 1.3. References [1] H. Bachmann, Y. Takeyama, K. Tasaka,
Cyclotomic analogues of finite multiplezeta values , preprint, 2017. arXiv:1707.05008.[2] D.M. Bradley,
Multiple q -zeta values , J. Algebra (2005), no. 2, 752–798.[3] L. Carlitz, A degenerate Staudt-Clausen theorem , Arch. Math. (Basel) (1956),28–33.[4] M. Kaneko and D. Zagier, Finite multiple zeta values , in preparation.[5] Y. Ohno and D. Zagier
Multiple zeta values of fixed weight, depth, and height ,Indag. Math. (N.S.), (4) (2001), 483–487.[6] J. Okuda and Y. Takeyama, On relations for the multiple q -zeta values , RamanujanJ. (2007), no. 3, 379–387.[7] Y. Takeyama, A q-analogue of non-strict multiple zeta values and basic hypergeo-metric series , Proc. Amer. Math. Soc. (2009), 2997–3002.[8] J. Zhao, q -Multiple zeta functions and q -multiple polylogarithms , Ramanujan J.14