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Báo cáo toán học: "Almost Product Evaluation of Hankel Determinants"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Almost Product Evaluation of Hankel Determinants. | Almost Product Evaluation of Hankel Determinants Omer Egecioglu Department of Computer Science University of California Santa Barbara CA 93106 omer@cs.ucsb.edu Timothy Redmond Stanford Medical Informatics Stanford University Stanford CA 94305 tredmond@stanford.edu Charles Ryavec College of Creative Studies University of California Santa Barbara CA 93106 ryavec@math.ucsb.edu Submitted Apr 25 2007 Accepted Dec 18 2007 Published Jan 1 2008 Mathematics Subject Classihcations 05A10 05A15 05A19 05E35 11C20 11B65 Abstract An extensive literature exists describing various techniques for the evaluation of Hankel determinants. The prevailing methods such as Dodgson condensation continued fraction expansion LU decomposition all produce product formulas when they are applicable. We mention the classic case of the Hankel determinants with binomial entries 3t 2 and those with entries 3 both of these classes of Hankel determinants have product form evaluations. The intermediate case 3k has not been evaluated. There is a good reason for this these latter determinants do not have product form evaluations. In this paper we evaluate the Hankel determinant of 3k . The evaluation is a sum of a small number of products an almost product. The method actually provides more and as applications we present the salient points for the evaluation of a number of other Hankel determinants with polynomial entries along with product and almost product form evaluations at special points. 1 Introduction A determinant Hn det aij 0 ij n THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R6 1 whose entries satisfy ai j ai j for some sequence ak k 0 is said to be a Hankel determinant. Thus Hn is the determinant of a special type of n 1 X n 1 symmetric matrix. In various cases of Hankel determinant evaluations special techniques such as Dodgson condensation continued fraction expansion and LU decomposition are applicable. These methods provide product formulas for a large class of Hankel determinants. A .