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),Georg

2、eAndrewsAbstract.Thepurposeofthisarticleisthreefold.First,itprovidesthereaderwithafewusefulandecienttoolswhichshouldenableher/himtoevaluatenontrivialde-terminantsforthecasesuchadeterminantshouldappearinher/hisresearch.Second,itlistsanumberofsuchdeterminantsthathavebe

3、enalreadyevaluated,togetherwithexplanationswhichtellinwhichcontextstheyhaveappeared.Third,itpointsoutreferenceswherefurthersuchdeterminantevaluationscanbefound.1.IntroductionImagine,youareworkingonaproblem.Asthingsdevelopitturnsoutthat,inordertosolveyourproblem,younee

4、dtoevaluateacertaindeterminant.Maybeyourdeterminantis1det;(1.1)1i;j;ni+jmath.CO/9902004v331May1999ora+bdet;(1.2)1i;jnai+joritispossibly+i+jdet;(1.3)0i;jn12ij1991MathematicsSubjectClassication.Primary05A19;Secondary05A1005A1505A1705A1805A3005E1005E

5、1511B6811B7311C2015A1533C4533D45.Keywordsandphrases.Determinants,Vandermondedeterminant,Cauchy'sdoublealternant,Pfaan,discreteWronskian,Hankeldeterminants,orthogonalpolynomials,Chebyshevpolynomials,Meixnerpolynomials,Meixner{Pollaczekpolynomials,Hermitepolynomials,Ch

6、arlierpolynomials,La-guerrepolynomials,Legendrepolynomials,ultrasphericalpolynomials,continuousHahnpolynomials,continuedfractions,binomialcoecient,Genocchinumbers,Bernoullinumbers,Stirlingnumbers,Bellnumbers,Eulernumbers,divideddierence,interpolation,planepartitions

7、,tableaux,rhombustilings,lozengetilings,alternatingsignmatrices,noncrossingpartitions,perfectmatchings,permutations,inversionnumber,majorindex,descentalgebra,noncommutativesymmetricfunctions.yResearchpartiallysupportedbytheAustrianScienceFoundationFWF,grantsP12094-MAT

8、andP13190-MAT.12C.KRATTENTHALERormaybex+y+jx+y+jdet:(1.4)1i;jnxi+2jx+i+2jHonestly,whichideaswouldyouhave?(Justtote