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Báo cáo toán học: "Elementary Proofs for Convolution Identities of Abel and Hagen–Rothe"

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:Elementary Proofs for Convolution Identities of Abel and Hagen–Rothe. | Elementary Proofs for Convolution Identities of Abel and Hagen-Rothe Wenchang Chu Hangzhou Normal University Institute of Combinatorial Mathematics Hangzhou 310036 P. R. China Submitted Feb 25 2010 Accepted Apr 20 2010 Published Apr 30 2010 Mathematics Subject Classifications 05A10 05A19 Abstract By means of series-rearrangements and finite differences elementary proofs are presented for the well-known convolution identities of Abel and Hagen-Rothe. 1 Introduction There are numerous identities in mathematical literature. Among them Newton s bino- mial theorem is well-known Ê n xk yn-k x n-k 0 Abel found its following deep generalization cf. Comtet 6 3.1 for example a a bk k c bk n k a c n a bk k n k n Another binomial identity is the Chu-Vandermonde convolution formula A Mt y V x A 2- lk n k n n n k 0 Email address chu.wenchang@unisalento.it THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N24 1 It has been generalized by Hagen and Rothe to the following one cf. Chu 3 4 Gould 8 and Graham et al 10 5.4 n t k 0 a a bk bk 2 rx c n k c 7 These convolution identities are fundamental in enumerative combinatorics. The reader can refer to Strehl 15 for a historical note. The existing proofs for the identities of Abel and Hagen-Rothe can be summarized as follows The classical Lagrange expansion formula Riordan 13 4.5 . Gould-Hsu Inverse series relations Chu and Hsu 1 5 . Generating function method Gould 8 9 see Chu 2 also . The Cauchy residue method of integral representation Egorychev 7 2.1 . Lattice path combinatorics Mohanty 11 4.2 and Narayana 12 Appendix . Riordan arrays which can trace back to Lagrange expansion Sprugnoli 14 . However to our knowledge there does not seem to have appeared really elementary proofs for these identities in classical combinatorics even though this has long been desirable. By utilizing the standard method of series-rearrangement that was systematically used by Wilf 16 this short paper will present elementary proofs for the convolution .