# calculus worked examples

May 12th, 2015
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ADVANCED DETERMINANT CALCULUSC. KRATTENTHALERInstitut fur Mathematik der Universitat Wien,Strudlhofgasse 4, A-1090 Wien, Austria.E-mail: kratt@pap.univie.ac.atWWW: http://radon.mat.univie.ac.at/People/krattDedicated to the pioneer of determinant evaluations (among many other things),George AndrewsAbstract. The purpose of this article is threefold. First, it provides the reader witha few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second,it lists a number of such determinants that have been already evaluated, together withexplanations which tell in which contexts they have appeared. Third, it points outreferences where further such determinant evaluations can be found.1. IntroductionImagine, you are working on a problem. As things develop it turns out that, inorder to solve your problem, you need to evaluate a certain determinant. Maybe yourdeterminant is1det,(1.1)1i,j,ni+jora+bdet,(1.2)1i,jnai+jor it is possiblydet0i,jn1+i+j2i j,(1.3)1991 Mathematics Subject Classification. Primary 05A19; Secondary 05A10 05A15 05A17 05A1805A30 05E10 05E15 11B68 11B73 11C20 15A15 33C45 33D45.Key words and phrases. Determinants, Vandermonde determinant, Cauchys double alternant,Pfaffian, discrete Wronskian, Hankel determinants, orthogonal polynomials, Chebyshev polynomials,Meixn

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