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@ 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 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. . 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
