Determinantal Evaluations Of Classical Combinatorial Sequences

By means of classical analytic method and combinatorial computational technique, this dissertation investigates some eigenvectors of tridiagonal matrices of Sylvester type, binomial determinantal formulae,generalizations of Cauchy and Vandermonde determinants as well as evaluations of determinants of Pascal matrices.The content is summarized as follows:1.Inspired by Askey[1]and Holtz[2],eigenvectors of the tridiagonal matrices of Sylvester type are explicitly determined by means of observation and experimentation through computer algebra,which are closely related to orthogonal polynomials named after Krawtchouk,(dual) Hahn and Racah as well as q-Racah polynomials.Then the rigorous demonstrations will be fulfilled through generating functions and manipulations on finite binomial sums.2.By means of partial fraction decomposition method,we evaluate a very general determinant of formal shifted factorial fractions,which covers numerous binomial determinantal identities,including several interesting ones appeared in Amdeberhan-Zeilberger [3].3.Applying Laplace expansion and divided differences,the author establishes a generalized Vandermonde determinant identity and several recurrence formulae of the extended Cauchy double alternants.Several special determinants are consequently computed.Furthermore,by means of finite difference expression,new proofs are presented for the determinants of Pascal matrices due to Krattenthaler[4]and Zakraj(?)ek -Petkov(?)ek[5].