A symbolic approach to multiple zeta values at the negative integers

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1 A symbolic approach to multiple zeta values at the egative itegers Victor H. Moll a, Li Jiu a Christophe Vigat a,b a Departmet of Mathematics, Tulae Uiversity, New Orleas, USA Correspodig author b LSS/Supelec, Uiversité Orsay Paris Sud, Frace Reçu le ***** ; accepté après révisio le Préseté par Résumé Ue approche symbolique des valeurs de la foctio zêta multiple aux etiers égatifs Nous utilisos des techiques de calcul symbolique afi d obteir ue expressio simple d ue cotiuatio aalytique de la foctio d Euler-Zagier, telle qu elle est proposée das la récete publicatio ]. Cette approche ous permet de calculer des idetités de recurrece sur les foctios géératrices aisi que sur la profodeur de la foctio zêta. Pour citer cet article : A. Nom, A. Nom, C. R. Acad. Sci. Paris, Ser. I Abstract Symbolic computatio techiques are used to derive some closed form expressios for a aalytic cotiuatio of the Euler-Zagier zeta fuctio evaluated at the egative itegers as recetly proposed i ]. This approach allows to compute explicitly some cotiguity idetities, recurreces o the depth of the zeta values ad geeratig fuctios. To cite this article: A. Nom, A. Nom, C. R. Acad. Sci. Paris, Ser. I Itroductio The multiple zeta fuctios, first itroduced by Euler ad geeralized by D. Zagier ], appear i diverse areas such as quatum field theory 5] ad ot theory 7]. These are defied by ζ r,..., r = 0<..., < < r r r addresses: vhm@tulae.edu Victor H. Moll, ljiu@tulae.edu Li Jiu, cvigat@tulae.edu Christophe Vigat. Preprit submitted to the Académie des scieces March 3, 05

2 where { i } are complex values, ad coverges whe the costraits Re r, ad Re r+ j, r, j= are satisfied see 8]. Their values at iteger poits =,..., r satisfyig are called multiple zeta values. A equivalet defiitio of these values is ζ r,..., r = >0,..., r> r. r The sum of the expoets + + r is called the weight of the zeta value, ad the umber r of these expoets is called its depth. Followig the result by Zhao 8] that the multiple zeta fuctio has a aalytic cotiuatio to the whole space C r, several authors have recetly proposed differet aalytic cotiuatios based o a variety of approaches: Aiyama et al. 3] used the Euler-Maclauri summatio formula ad Matsumoto 4] the Melli-Bares itegral formula. B. Sadaoui ] provided recetly such aalytic cotiuatio based o Raabe s idetity, which lis the multiple itegral Y a =,+ r to the multiple zeta fuctio Z,z = by,..., r dx x + a x + a + x + a... x + a + + x r + a r r + z + z + + z... + z + + r + z r r Y 0 = Z, z dz. 0,] r B. Sadaoui uses a classical iversio argumet to obtai a aalytic cotiuatio of the multiple zeta fuctio defied at egative iteger argumets =,..., r. The argumet uses the followig three steps: - the itegral Y a is computed for values of,..., r that satisfy the covergece coditios, - the values are replaced by i this result: it is the show that Y a is a polyomial i the variable a, - the variables a = a,..., a r are replaced by B,..., B r, ad each Beroulli symbol B satisfies the two evaluatio rules: evaluatio rule : each power B p of the Beroulli symbol B should be evaluated as B p B p, 3 the p th Beroulli umber evaluatio rule : for two differet symbols B ad B l, l, the product B p Bq l is evaluated as B p Bq l B pb q, 4 the product of the Beroulli umbers B p ad B q. If = l, the first rule applies to give the evaluatio B p Bq B p+q.

3 Example. A example of depth, appearig i ], is ow computed usig the rules above. The itegral Y a, is explicitly computed ad, replacig, by, gives Y a,a, = l l a l al. =0 =0 The substitutig the variables a ad a with the Beroulli symbols B ad B gives ζ, = l l B l B l =0 l =0 l =0 l =0 Usig the evaluatio rules 3 ad 4 for the Beroulli symbols, the multiple zeta value of depth at, is ζ, = l l B l B l l =0 l =0 l =0 The geeral case is give i, eq. 4.0] as the r fold sum ζ r,..., r = r + r,..., r j= + r... l,...,l r l l r r where,..., r 0, l j j for j r ad l + r + ad = r j, = j= r i+r j+ i=j i=j+ i j i=j i + r j + B l... B lr l r i=j+ i 5 r j. 6 A symbolic expressio for 5 is proposed here. This is used as a coveiet tool to derive some specific zeta values at egative itegers, cotiguity idetities for the multiple zeta fuctios, recursios o their depth ad geeratig fuctios. j=. Mai result Itroduce first the symbols C,,..., defied recursively i terms of the Beroulli symbols B,..., B r as C = B, C, = C + B,... ad C,,...,+ = C,,..., + B + with the symbolic computatio rule: C symbols rule: All symbols C,,..., are expaded usig the above idetities to express them oly i terms of B. The evaluatio rules 3 ad 4 for the Beroulli symbols are the applied. Example. For example,. This corrects a typo i, eq. 4.0] 3

4 C C = C is evaluated as C + B = C + B = B+ + B. B + + B The ext result is give i terms of this otatio. Theorem. The multiple zeta values 5 at the egative itegers,..., r are give by ζ r,..., r = C +,...,. 7 Proof. The ier sum i 5, i its Beroulli symbols versio, + r r... B l... B lr, ca be summed to The classical idetity It follows that ζ r = r + Summig first over gives ζ r = r + l,...,l r l l = + B +r + B... + B r r. for Beroulli symbols B + = B, with defied i 6 reduces this to C +r B... Br r,..., r l r + B +r B... Br r. 8 j= C + C + +r+r B Br r 3,..., r r i+r j+ i=j i=j+ i j r i=j i + r j +. i=j+ i j=3 r i+r j+ i=j i=j+ i j r i=j i + r j +. i=j+ i The result ow follows by summig, i order, over the remaiig idices. Observe that the reductio 8 performed i the proof allows to restate a simpler versio of Sadaoui s formula 5 as the more tractable r fold sum ζ r,..., r = + r,..., r j= r i+r j+ i=j i=j+ i Bl... B lr j r i=j i + r j +. 9 i=j+ i Observe moreover that the derivatio of 7 is uchaged if the symbols B,..., B r are replaced by a geeralizatio of the Beroulli symbol B, amely the polyomial Beroulli symbol B + z defied by B + z = B z,. this idetity ca be deduced from the geeratig fuctio exp zb = z exp z. 4

5 the Beroulli polyomial of degree. The same proof as above yields the ext statemet. Theorem.3 The aalytic cotiuatio of the zeta fuctio as give i ] ca be writte as ζ r,..., r, z,..., z r = C i+,...,i z,..., z i 0 with ad C z = z + B = B z i=, C, z, z = C z + B + z,... C,,...,+ z,..., z + = C,,..., z,..., z + B + + z A geeral recursio formula o the depth The methods above are ow used to produce a geeral recursio formula o the depth of the zeta fuctio. Theorem 3. The multiple zeta fuctios satisfy the recursio rule ζ r ; z = r r + r+ r + Itroducig the ew zeta symbol Z r with the evaluatio rule 3 this recursio rule ca be writte symbolically as ζ r,..., r ; z B r+ z r. Z r = ζ r,..., r, r ; z, ζ r ; z = B Z r r+ r = ζ r ; Z r. r + Proof 3. Start from 0 ad expad the last term C r+,...,r z,..., z r = by usig the biomial formula to produce ζ r,..., r, z,..., z r = r r+ The idetify as to obtai the desired result. 3. ote that Z 0 r r i= C r +,...,r z,..., z r + B r z r r+ r + r r + i= C r++,...,r z,..., z r B r+ r z r. C i+,...,i z,..., z i r+ C i+,...,i z,..., z i C r++,...,r z,..., z r + +r +r + ζ r,..., r, r ; z 5

6 Usig the symbol Z, this idetity ca be rewritte as ζ r,..., r, z,..., z r = r r+ B Z r r+ ad the iitial value provides the stated recursio. ζ ; z = z + B Cotiguity idetities The multiple zeta fuctio at egative iteger values satisfies cotiguity idetities i the z variables. Two of them are preseted here. Theorem 4. The zeta fuctio satisfies the cotiguity idetity ζ r,..., r ; z,..., z r, z r + = ζ r,..., r ; z,..., z r, z r + r z r Z r r. Example 3. I the case of the zeta fuctio of depth, ζ,, z, z + = ζ,, z, z + + z Z ad the secod term is expaded as + z ζ ; z. Proof 4. Expad ζ r,..., r ; z,..., z r, z r + = ad use the idetity o Beroulli polyomials r + C+ z... C r +,...,r z,..., z r r + r+ B r+ z r + = B r+ z r + r + z r r C r ++,...,r z,..., z r B r+ z r +. to produce the result. The correspodig result for a shift i the first variable admits a similar proof. Theorem 4.3 The depth- zeta fuctio satisfies the cotiguity idetities ζ,, z +, z = ζ,, z, z z B + z + z. 6

7 5. A Geeratig Fuctio The geeratig fuctio of the zeta values at egative itegers is defied by F r w,..., w r =,..., r 0 w... wr r!... r! ζ r,..., r. 3 A recurrece for F r is preseted below. The iitial coditio is give i terms of the geeratig fuctio for Beroulli umbers F B w = + =0 B! z = w e w. Theorem 5. The geeratig fuctio of the zeta values at egative itegers satisfies the recurrece F r w,..., w r = w r F r w,..., w r F B w r F r w,..., w r, w r + w r ] with the iitial value F w = w e w B ] = F B w w. Moreover, the represetatio of the shift operator as exp a w f w = f w + a ad F w, z = e wb+z ] give the recursio symbolically as w so that F r w,..., w r = F w r, Proof 5. Start from F r w,..., w r = ad expad C,...,r e wrc,...,r = F r w,..., w r = F w r,,... r w w r wr r! r! w r =0 to deduce that F r w,..., w r is w r! w r F w r, F r w,..., w r, w r + +r C + C r+,...,r = F w, F w. w C,...,j e wjc,...,j, j= + + C,...,r + B r + = e wrc,...,r +Br, w r r C,...,j e wjc,...,j r C,...,j e wjc,...,j e wrbr C,...,r e wr +wrc,...,r w r j= = w r F r w,..., w r w r F B w r F r w,..., w r, w r + w r. j= 7

8 6. Shuffle Idetity Multiple zeta values at positive itegers satisfy shuffle idetities, such as ζ, + ζ, + ζ + = ζ ζ. The aalytic cotiuatio techique used i ] does ot preserve this idetity at egative itegers, while others do for example, see 6]. The followig theorem gives the correctio terms. Theorem 6. The zeta values at egative itegers as defied i ] satisfy the idetity ζ, + ζ, + ζ ζ ζ = +!! + +! B Remar. Whe + is odd, B + + = 0 so that the shuffle idetity 4 holds for ζ, as expected, sice the depth zeta fuctio is holomorphic at these poits. Proof 6. Let δ w, w = F w, w + F w, w + F w + w F w F w. A elemetary calculatio gives w δ w, w = + w coth w coth w. w + w The expasios w coth w + w + = +! B + ad = w w + w w w ow produce δ w, w = +,l=0 l 0 l B + w +l+ w l + w l +! w l. Idetifyig the coefficiet of w w i this series expasio gives the result. 7. Specific multiple zeta values This fial sectio gives some examples of the evaluatio at egative itegers of the zeta fuctio, obtaied from 5 ad. : for depth r =, ad : for depth r = 3, ζ, 0 = B+ + ] B +, 5 + ζ 0, = + + B + + B + ].; 6 ζ 3, 0, 0 = B B B + + ] 7

9 ad ζ 3 0,, 0 = B + B ] B : as a fial example, the recursio rule is used to compute the value ζ 3 0, 0, as ζ 3 0, 0, = B Z 3 = B 3 Z 0 3B Z + 3BZ Z = 3 B 3ζ 0, 0 3B ζ 0, + 3B ζ 0, ζ 0, 3 = 60. Acowledgemets The first author acowledges the partial support of NSF-DMS The secod author is partially supported, as a graduate studet, by the same grat. The wor of the third author was partially fuded by the icode Istitute, a research project of the Idex Paris-Saclay. Refereces ] B. Sadaoui, Multiple zeta values at the o-positive itegers, C. R. Acad. Sci. Paris, Ser., 04, -35, ] D. Zagier, Values of zeta fuctios ad their applicatios, Progress i Math., 0, 994, ] S.Aiyama S.Egami ad Y.Taigawa, A aalytic cotiuatio of multiple zeta fuctios ad their values at o-positive itegers, Acta Arith. XCVIII, 00, ] K. Matsumoto, The aalytic cotiuatio ad the asymptotic behavior of certai multiple zeta-fuctios I, J. Number Theory 0, 003, ] D. J. Broadhurst, Exploitig the,440-fold symmetry of the master two-loop diagram, Zeit. Phys. C 3, 986, ] D. Macho ad S. Paycha, Nested sums of symbols ad reormalised multiple zeta fuctios, preprit, arxiv:math/ ] T. Taamui, The Kotsevich ivariat ad relatios of multiple zeta values, Kobe J. Math. 6, 999, ] Zhao, J.: Aalytic cotiuatio of multiple zeta-fuctios. Proc. Am. Math. Soc. 8, 000,