tailieunhanh - Báo cáo toán học: "Ap´ry’s Double Sum is Plain Sailing Indeed e"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Ap´ry’s Double Sum is Plain Sailing Indeed e. | Apery s Double Sum is Plain Sailing Indeed Carsten Schneider Research Institute for Symbolic Computation J. Kepler University Linz A-4040 Linz Austria Submitted Dec 12 2006 Accepted Jan 19 2007 Published Jan 29 2007 Mathematics Subject Classification 65B10 33F10 68W30 Abstract We demonstrate that also the second sum involved in Apery s proof of the irrationality of 3 becomes trivial by symbolic summation. In his beautiful survey 4 van der Poorten explained that Apery s proof 1 of the irrationality of 3 relies on the following fact If A fn A 2 Ỉ A 2 a n X k kj k 0 and b n X n k i2 fn i2 H3 X . m 1 k k I n 2m3fn mWnm 1 k 0 X m 1 2m V m W where Hn3 pn 1 i3 are the harmonic numbers of order three then both sums a n and b n satisfy the same recurrence relation n 1 3A n - 2n 3 17n2 51n 39 A n 1 n 2 3A n 2 0. 2 Van der Poorten points out that Henri Cohen and Don Zagier showed this key ingredient by some rather complicated but ingenious explanations 4 Section 8 based on the creative telescoping method. Due to Doron Zeilberger s algorithmic breakthrough 9 the a n -case became a trivial exercise. Also the b n -case can be handled by skillful application of computer algebra In 10 Zeilberger was able to generalize the Zagier Cohen method in the setting of Supported by the SFB-grant F1305 and the grant P16613-N12 of the Austrian FWF THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N5 1 WZ-forms. Later developments for multiple sums 8 7 together with holonomic closure properties 5 3 enable alternative computer proofs of the b n -case see . 2 . Nowadays also the b n -case is completely trivialized Using the summation package Sigma 6 we get plain sailing - instead of plane sailing cf. van der Poorten s statement in 4 Section 8 . Namely after loading the package into the computer algebra system Mathematica In i Sigma - A summation package by Carsten Schneider RISC-Linz we insert our sum mySum b n J fn k In 2 mySum y I k 0 k k H 3 X m