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Báo cáo toán học: "Hayman admissible functions in several variables"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Hayman admissible functions in several variables. | Hayman admissible functions in several variables Bernhard Gittenberger and Johannes Mandlburger Institute of Discrete Mathematics and Geometry Technical University of Vienna Wiedner Hauptstrafie 8-10 104 A-1040 Wien Austria gittenberger@dmg.tuwien.ac.at Submitted Sep 12 2006 Accepted Nov 1 2006 Published Nov 17 2006 Mathematics Subject Classifications 05A16 32A05 Abstract An alternative generalisation of Hayman s concept of admissible functions to functions in several variables is developed and a multivariate asymptotic expansion for the coefficients is proved. In contrast to existing generalisations of Hayman admissibility most of the closure properties which are satisfied by Hayman s admissible functions can be shown to hold for this class of functions as well. 1 Introduction 1.1 General Remarks and History Hayman 20 defined a class of analytic functions y ynxn for which their coefficients yn can be computed asymptotically by applying the saddle point method in a rather uniform fashion. Moreover those functions satisfy nice algebraic closure properties which makes checking a function for admissibility amenable to a computer. Many extensions of this concept can be found in the literature. E.g. Harris and Schoenfeld 19 introduced an admissibility imposing much stronger technical requirements on the functions. The consequence is that they obtain a full asymptotic expansion for the coefficients and not only the main term. The disadvantage is the loss of the closure properties. Moreover it can be shown that if y x is H-admissible then ey x is HS-admissible see 37 and the error term is bounded. There are numerous applications of H-admissible or HS-admissible functions in various fields see for instance 1 2 3 8 9 10 11 13 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 . This research has been supported by the Austrian Science Foundation FWF grant P16053-N05 as well as grant S9604 part of the Austrian Research Network Analytic Combinatorics and Probabilistic