sum relations for multiple zeta values and connection formulas for the gauss hypergeometric

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1、SumrelationsformultiplezetavaluesandconnectionformulasfortheGausshypergeometricfunctionsTakashiAoki∗andYasuoOhno†AbstractWegiveanexplicitrepresentationforthesumsofmultiplezeta-starvaluesoffixedweightandheightintermsofRiemannzetavalues.1IntroductionInthisarticleweestablishanewfamilyofrelation

2、sbetweensumsofmultiplezetavaluesandRiemannzetavalues.Thisfamilycontainsrelationswhichdonotappearinthefamilyofrelationsgivenin[6],[9],and[11].Concerningmultiplezetavalues,therearetwotypesofdefinition:multi-plezetavalues“withoutequality”andones“withequality”(seebelow).Theformerismainlyusedinma

3、thematicalliteratureandthelatteristhemainsubjectofthisarticle.Normallymultiplezetavalues(MZVsforshort)meantheformerandaredenotedbyζ(k).Wetentativelycallthelattermultiplezeta-starvaluesanddenotethembyζ∗(k)todistinguishthemfromordinaryarXiv:math/0307264v1[math.NT]19Jul2003ones.Weabbreviatethe

4、mtoMZSVs.Theyareclassicobjectsalthoughtherehadbeennonameofthem.Infact,EulerwasthefirstmathematicianwhowasinterestedinmultiplezetavaluesandhemainlytreatedMZSVs[3].Recently,Hoffman[5]pointedoutthesignificanceofconsideringMZSVsaswellasMZVs.ThenotationSwasusedthereforζ∗.Themainresultofthisarticles

5、howsthatthesumofMZSVswithfixedweightandheightturnsouttobearationalmultipleofRiemannzetavalueat∗SuppotedinpartbyJSPSGrant-in-AidNo.14340042andbyNo.15540190†SuppotedinpartbyJSPSGrant-in-AidNo.15740025andbyNo.155401901theweight.ConsideringMZSVsclarifiestheimportanceofthosetwoindices:weightandhei

6、ght.Theyhavebeenplayedarolein[8],[11].TheemploymentoftheindicesandMZSVsisaneatwaytoformulatesystematicdescriptionofrelationsthatholdamongMZVs.Anotherimportantindexisdepth.WebelievethatMZSVsandthethreeindices:weight,heightanddepthwillplayanimportantroleininvestigationofthestructureofQ-algebr

7、ageneratedbyMZVs.(NotethatthisalgebracoincideswithQ-algebrageneratedbyMZSVs.)Aninterestingfeatureofthemethodemployedinourproofisrelatedtothetheoryofdifferentialequationsinthecomplexdomain.Themethodisavariationon[11]andtheuseofconnectionformulasoftheGaussh