ADVANCED DETERMINANT CALCULUS
math.CO/9902004 v3 31 May 1999
C. KRATTENTHALER†
Institut f¨ ur Mathematik der Universit¨at Wien, Strudlhofgasse 4, A-1090 Wien, Austria. E-mail:
[email protected] WWW: http://radon.mat.univie.ac.at/People/kratt Dedicated to the pioneer of determinant evaluations (among many other things), George Andrews Abstract. The purpose of this article is threefold. First, it provides the reader with a 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 with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.
1. Introduction Imagine, you are working on a problem. As things develop it turns out that, in order to solve your problem, you need to evaluate a certain determinant. Maybe your determinant is 1 , (1.1) det 1≤i,j,≤n i+j or a+b det , (1.2) 1≤i,j≤n a−i+j or it is possibly
det
0≤i,j≤n−1
µ+i+j 2i − j
,
(1.3)
1991 Mathematics Subject Classification. Primary 05A19; Secondary 05A10 05A15 05A17 05A18 05A30 05E10 05E15 11B68 11B73 11C20 15A15 33C45 33D45. Key words and phrases. Determinants, Vandermonde determinant, Cauchy’s double alternant, Pfaffian, discrete Wronskian, Hankel determinants, orthogonal polynomials, Chebyshev polynomials, Meixner polynomials, Meixner–Pollaczek polynomials, Hermite polynomials, Charlier polynomials, Laguerre polynomials, Legendre polynomials, ultraspherical polynomials, continuous Hahn polynomials, continued fractions, binom