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Báo cáo toán học: "On the non-holonomic character of logarithms, powers, and the nth prime function"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: On the non-holonomic character of logarithms, powers, and the nth prime function. | On the non-holonomic character of logarithms powers and the nth prime function Philippe Flajolet Algorithms Project INRIA Rocquencourt F-78153 Le Chesnay France Philippe.Flajolet AT inria.fr Stefan Gerhold Research Institute for Symbolic Computation Johannes Kepler University Linz Austria stefan.gerhold AT risc.uni-linz.ac.at Bruno Salvy Algorithms Project INRIA Rocquencourt F-78153 Le Chesnay France Bruno.Salvy AT inria.fr Submitted Jan 21 2005 Accepted Mar 30 2005 Published Apr 28 2005 Mathematics Subject Classifications 05A15 11B83 33E30 Es ist eine Tatsache daB die genauere Kenntnis des Verhaltens einer analytischen Funktion in der Nahe ihrer singularen Stellen eine Quelle von arithmetischen Satzen ist. 1 Erich Hegke 27 Kap. VIII Abstract We establish that the sequences formed by logarithms and by fractional powers of integers as well as the sequence of prime numbers are non-holonomic thereby answering three open problems of Gerhold El. J. Comb. 11 2004 R87 . Our proofs depend on basic complex analysis namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem. A brief discussion is offered regarding the scope of singularity-based methods and several naturally occurring sequences are proved to be non-holonomic. Supported in part by the SFB-grant F1305 of the Austrian FWF 1 It is a fact that the precise knowledge of the behaviour of an analytic function in the vicinity of its singular points is a source of arithmetic properties. the electronic journal of combinatorics 11 2 2005 A2 1 Introduction A sequence fn n 0 of complex numbers is said to be holonomic or P-recursive if it satisfies a linear recurrence with coefficients that are polynomial in the index n that is Po n fn d Pi n fn d-i - Pd n fn 0 n 0 1 for some polynomialsPj X G C X . A formal power series f z J2n 0 fnzn is holonomic or d-finite if it satisfies a linear differential equation with coefficients that are polynomial in .