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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: Combinatorial interpretations of the Jacobi-Stirling numbers. | Combinatorial interpretations of the Jacobi-Stirling numbers Yoann Gelineau and Jiang Zeng Universite de Lyon Universite Lyon 1 Institut Camille Jordan UMR 5208 du CNRS F-69622 Villeurbanne Cedex France gelineau@math.univ-lyon1.fr zeng@math.univ-lyon1.fr Submitted Sep 24 2009 Accepted May 4 2010 Published May 14 2010 Mathematics Subject Classifications 05A05 05A15 33C45 05A10 05A18 34B24 Abstract The Jacobi-Stirling numbers of the first and second kinds were introduced in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds. 1 Introduction It is well known that Jacobi polynomials pha t satisfy the classical second-order Jacobi differential equation 1 t2 y t p a a p 2 t y t n n a p 1 y t 0. 1.1 Let a p y t be the Jacobi differential operator l -ri M í 1 . ty - 1 Í a I 1 í 3 Iy í Then equation 1.1 is equivalent to say that y l t is a solution of a 0 y t n n a p 1 y t . the electronic journal of combinatorics 17 2010 R70 1 Table 1 The first values of JSn z k n 1 2 3 4 5 6 1 1 z 1 z 1 2 z 1 3 z 1 z 1 5 2 1 5 3z 21 24z 7z2 85 141z 79z2 15z3 341 738z 604z2 222z3 31z4 3 1 14 6z 147 120z 25z2 1408 1662z 664z2 90z3 4 1 30 10z 627 400z 65z2 5 1 55 15z 6 1 In 5 Theorem 4.2 for each n G N Everitt et al. gave the following expansion of the n-th composite power of a p 1 _ b a 1 1 4fn r t i nk fp 4 sk 1 - b a k 1 I SI k t k . 1 b 1 b a g y b A 1 ự sn 1 b 1 b y b where P a sn are called the Jacobi-Stirling numbers of the second kind. They 5 4.4 also gave an explicit summation formula for P a sn numbers showing that these .
