Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: An Ap´ry-like difference equation e for Catalan’s constant | An Apery-like difference equation for Catalan s constant W. Zudilin Moscow Lomonosov State University Department of Mechanics and Mathematics Vorobiovy Gory GSP-2 Moscow 119992 RUSSIA URL http E-mail address wadim@ Submitted Jan 18 2002 Accepted Mar 31 2003 Published Apr 23 2003 MR Subject Classifications Primary 11J70 11Y60 33F10 Secondary 11B37 11B65 11M06 33C20 33C60 39A05. Abstract Applying Zeilberger s algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan s constant with rational coefficients we obtain a second-order difference equation for these forms and their coefficients. As a consequence we derive a new way of fast calculation of Catalan s constant as well as a new continued-fraction expansion for it. Similar arguments are put forward to deduce a second-order difference equation and a new continued fraction for z 4 n4 90. 1 Introduction One of the most crucial and quite mysterious ingredients in Apery s proof 1 8 of the irrationality of z 2 and z 3 is the existence of the difference equations n 1 2U 1 11n2 11n 3 un nZ Un-1 0 u0 1 u1 3 v 0 v 1 5 1 and n 1 3un 1 2n 1 17n2 17n 5 un n3un 0 U0 1 u 1 5 v0 0 v 1 6 with the following properties of their solutions Am Z7 z 2 Am s z 3 . n u n u1 cn n THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 R14 1 Unexpected inclusions u n D vn E Z and u n Dnv n E Z where Dn denotes the least common multiple of the numbers 1 2 . n and D0 1 for completeness together with the prime number theorem Dnn - e as n - to and Poincare s theorem then yield the following asymptotic behaviour of the linear forms Dnu nC 2 Dnv n and Dnu nz 3 Dnv n with integral coefficients lirn Dnu nZ 2 Dnv n 1n 5 - 1 n 2 lim Duz 3 Divn1 e V2 1 4 1 n and thus one obtains that both z 2 and z 3 are irrational. The two following decades after 1 were full of attempts to indicate the total list of the second-order recursions with integral solutions and
