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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: On Some Non-Holonomic Sequences. | On Some Non-Holonomic Sequences Stefan Gerhold Research Institute for Symbolic Computation Johannes Kepler University Linz Austria stefan.gerhold@risc.uni-linz.ac.at Submitted Oct 15 2003 Accepted Nov 25 2004 Published Dec 7 2004 Mathematics Subject Classifications 11B37 11R32 11J81 Abstract A sequence of complex numbers is holonomic if it satisfies a linear recurrence with polynomial coefficients. A power series is holonomic if it satisfies a linear differential equation with polynomial coefficients which is equivalent to its coefficient sequence being holonomic. It is well known that all algebraic power series are holonomic. We show that the analogous statement for sequences is false by proving that the sequence ựngn is not holonomic. In addition we show that nn n the Lambert W function and log n n are not holonomic where in the case of log n n we have to rely on an open conjecture from transcendental number theory. 1 Introduction A sequence u N C is called holonomic P-recursive P-finite over a field K c C if it satisfies a homogeneous linear recurrence p0 n u n p1 n u n 1 . pd n u n d 0 n 0 1 where the pk are polynomials with coefficients in K and pd is not identically zero. If K is not mentioned it is understood to be C. Many combinatorial sequences are holonomic. A formal power series f z J2ra 0 u n zn is holonomic D-finite P-finite if it satisfies a homogeneous linear ordinary differential equation po z f z pi z f z . pd z f d z 0 2 with polynomial coefficients. Holonomicity of meromorphic functions is defined in the same way. It is well known 8 that a power series is holonomic if and only if its coefficient sequence is. Supported by the SFB-grant F1305 of the Austrian FWF THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R87 1 There are powerful methods for showing that certain power series are not holonomic. For instance given that f is holonomic 1 f if dehned is holonomic if and only if f0 f is algebraic and exp J f is holonomic if and only if f is algebraic