advanced_determinant_calculus

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1、ADVANCEDDETERMINANTCALCULUSC.KRATTENTHALERyInstitutf•urMathematikderUniversit•atWien,Strudlhofgasse4,A-1090Wien,Austria.E-mail:kratt@pap.univie.ac.atWWW:http://radon.mat.univie.ac.at/People/krattDedicatedtothepioneerofdeterminantevaluations(amongmanyotherthings),GeorgeAndrews

2、Abstract.Thepurposeofthisarticleisthreefold.First,itprovidesthereaderwithafewusefulandecienttoolswhichshouldenableher/himtoevaluatenontrivialde-terminantsforthecasesuchadeterminantshouldappearinher/hisresearch.Second,itlistsanumberofsuchdeterminantsthathavebeenalreadyevaluat

3、ed,togetherwithexplanationswhichtellinwhichcontextstheyhaveappeared.Third,itpointsoutreferenceswherefurthersuchdeterminantevaluationscanbefound.1.IntroductionImagine,youareworkingonaproblem.Asthingsdevelopitturnsoutthat,inordertosolveyourproblem,youneedtoevaluateacertaindeter

4、minant.Maybeyourdeterminantis1det;(1.1)1i;j;ni+jmath.CO/9902004v331May1999ora+bdet;(1.2)1i;jnai+joritispossibly+i+jdet;(1.3)0i;jn12ij1991MathematicsSubjectClassication.Primary05A19;Secondary05A1005A1505A1705A1805A3005E1005E1511B6811B7311C2015A1533C4533D45

5、.Keywordsandphrases.Determinants,Vandermondedeterminant,Cauchy'sdoublealternant,Pfaan,discreteWronskian,Hankeldeterminants,orthogonalpolynomials,Chebyshevpolynomials,Meixnerpolynomials,Meixner{Pollaczekpolynomials,Hermitepolynomials,Charlierpolynomials,La-guerrepolynomials,L

6、egendrepolynomials,ultrasphericalpolynomials,continuousHahnpolynomials,continuedfractions,binomialcoecient,Genocchinumbers,Bernoullinumbers,Stirlingnumbers,Bellnumbers,Eulernumbers,divideddierence,interpolation,planepartitions,tableaux,rhombustilings,lozengetilings,alternat

7、ingsignmatrices,noncrossingpartitions,perfectmatchings,permutations,inversionnumber,majorindex,descentalgebra,noncommutativesymmetricfunctions.yResearchpartiallysupportedbytheAustrianScienceFoundationFWF,grantsP12094-MATandP13190-MAT.12C.KRATTENTHALERormaybex+y+jx+y+jde

8、t:(1.4)1i;jnxi+2jx+i+2jHonestly,whichideaswouldyouhave?(Justtote