Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài:On a combinatorial problem of Asmus Schmidt. | On a combinatorial problem of Asmus Schmidt W. Zudilin Department of Mechanics and Mathematics Moscow Lomonosov State University Vorobiovy Gory GSP-2 Moscow 119992 RUSSIA URL http E-mail address wadim@ Submitted Dec 29 2003 Accepted Feb 26 2004 Published Mar 9 2004. MR Subject Classifications 11B65 33C20. Abstract For any integer r 2 define a sequence of numbers c k o . independent of the parameter n by En n r n k r n n n k r .w V k UK k Ck k 0 k 0 n 0 1 2 . . TIT 1 11 1 1 r . We prove that all the numbers Ck are integers. 1 Stating the problem The following curious problem was stated by A. L. Schmidt in 5 in 1992. Problem 1. For any integer r 2 define a sequence of numbers ckr k o i . independent of the parameter n by n 0 1 2 . 1 Is it then true that all the numbers c r are integers The work is supported by an Alexander von Humboldt research fellowship and partially supported by grant no. 03-01-00359 of the Russian Foundation for Basic Research. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R22 1 An affirmative answer for r 2 was given in 1992 but published a little bit later independently by Schmidt himself 6 and by V. Strehl 7 . They both proved the following explicit expression n 3 2 cn 2- j 0 n 0 1 2 . 2 j which was observed experimentally by W. Deuber W. Thumser and B. Voigt. In fact Strehl used in 7 the corresponding identity as a model for demonstrating various proof techniques for binomial identities. He also proved an explicit expression for the sequence cn3 thus answering Problem 1 affirmatively in the case r 3. But for this case Strehl had only one proof based on Zeilberger s algorithm of creative telescoping. Problem 1 was restated in 3 Exercise 114 on p. 256 with an indication on p. 549 that H. Wilf had shown the desired integrality of cd for any r but only for any n 9. We recall that the first non-trivial case r 2 is deeply related to the famous Apery n 2 n 2 numbers k fcj k the denominators of rational .