Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài:Lindel¨f Representations and (Non-)Holonomic o Sequences. | Lindelof Representations and Non- Holonomic Sequences Philippe Flajolet Algorithms Project INRIA Rocquencourt F-78153 Le Chesnay France AT Stefan Gerhold TU Vienna Austria and Microsoft Research-INRIA Orsay France sgerhold AT Bruno Salvy Algorithms Project INRIA Rocquencourt F-78153 Le Chesnay France AT Submitted Jun 9 2009 Accepted Dec 9 2009 Published Jan 5 2010 Mathematics Subject Classifications 11B83 30E20 33E20 Keywords Holonomic sequence D-finite function Lindelof representation Abstract Various sequences that possess explicit analytic expressions can be analysed asymptotically through integral representations due to Lindeloof which belong to an attractive but somewhat neglected chapter of complex analysis. One of the outcomes of such analyses concerns the non-existence of linear recurrences with polynomial coefficients annihilating these sequences and accordingly the non-existence of linear differential equations with polynomial coefficients annihilating their generating functions. In particular the corresponding generating functions are transcendental. Asymptotic estimates of certain finite difference sequences come out as a byproduct of the Lindeloof approach. Introduction There has been recently a surge of interest in methods for proving that certain sequences coming from analysis or combinatorics are non-holonomic. Recall that a sequence fn This work was partially supported by CDG BA-CA AFFA and the joint INRIA-Microsoft Research Laboratory. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R3 1 is holonomic or P-recursive if it satisfies a linear recurrence with coefficients that are polynomial equivalently rational in the index n that is d Pk n fn-k 0 Pk n G C n P0 0. k 0 Put otherwise its generating function f z called holonomic or D-finite satisfies a linear differential equation with coefficients that are polynomial equivalently rational in the variable z that is with f z Vn . fnzn y dk
