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: On a new family of generalized Stirling and Bell numbers. | On a new family of generalized Stirling and Bell numbers Toufik Mansour Department of Mathematics University of Haifa 31905 Haifa Israel toufik@ Matthias Schork Camillo-Sitte-Weg 25 60488 Frakfurt Germany mschork@ Mark Shattuck Department of Mathematics University of Haifa 31905 Haifa Israel maarkons@ Submitted Feb 11 2011 Accepted Mar 24 2011 Published Mar 31 2011 Mathematics Subject Classification 05A15 05A18 05A19 11B37 11B73 11B75 Abstract A new family of generalized Stirling and Bell numbers is introduced by considering powers VU n of the noncommuting variables U V satisfying UV VU hVs. The case s 0 and h 1 corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers the recursion relation is given and explicit expressions are derived. Furthermore they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as s-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore an analogue of Dobinski s formula is given for these Bell numbers. 1 Introduction The Stirling numbers of first and second kind are certainly among the most important combinatorial numbers as can be seen from their occurrence in many different contexts see . 6 14 35 38 42 and the references given therein. One of these interpretations is THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P77 1 in terms of normal ordering special words in the Weyl algebra generated by the variables U V satisfying UV - VU 1 1 where on the right-hand side the identity is denoted by 1. A concrete representation for 1 is given by the operators U D d V X dx acting on a suitable space of functions where X f x xf x . In the mathematical literature the simplification
