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...
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Báo cáo toán học: "Almost Product Evaluation of Hankel Determinants" Almost Product Evaluation of Hankel Determinants ¨ Omer E˘ecio˘lu g g 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 Classifications: 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 3kk and those with entries 3k ; both of these classes of Hankel +2 k determinants have product form evaluations. The intermediate case, 3kk has not +1 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 +1 . The evaluation is a sum of a small number of products, an almost product. k 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 IntroductionA determinant Hn = det[ai,j ]0≤i,j ≤n 1the electronic journal of combinatorics 15 (2008), #R6whose entries satisfy ai,j = ai+jfor some sequence {ak }k≥0 is said to be a Hankel determinant. Thus Hn is the determinantof a special type of (n + 1) × (n + 1) symmetric matrix. In various cases of Hankel determinant evaluations, special techniques such as Dodgsoncondensation, continued fraction expansion, and LU decomposition are applicable. Thesemethods provide product formulas for a large class of Hankel determinants. A moderntreatment of the theory of determinant evaluation including Hankel determinants as wellas a substantial bibliography can be found in Krattenthaler [7, 8]. The product form determinants of special note are those whose factors have someparticular attraction. Factorials and other familiar combinatorial entities that appear asfactors have an especially pleasing quality, and we find an extensive literature devoted tothe evaluation of classes of Hankel determinants as such products. Several classical Hankel determinants involve entries that are binomial coefficients orexpressions closely related to binomial coefficients. Perhaps the most well-known of theseis where ak = 2kk , and ak = 2k1 2kk , for which Hn = 1 for all n. The binomial +1 +1 +1entries 3k 3k + 2 ak = , ak = (1) k kalso yield product evaluations for the corresponding Hn as we give in (4) and (3). In fact,product formulas have been shown to exist for a host of other cases (see Gessel and Xin[4]), and we only mention 1 3k + 1 9k + 14 3k + 2 ak = , ak = 3k + 1 k (3k + 4)(3k + 5) k + 1as representatives. However, within the restricted class of Hankel determinants defined by the binomialcoefficients βk + α (β,α) ak = a k = , (2) kparametriz ...