TAILIEUCHUNG - Báo cáo toán học: "Hybrid Proofs of the q-Binomial Theorem and other identities"

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: Hybrid Proofs of the q-Binomial Theorem and other identities. | Hybrid Proofs of the q-Binomial Theorem and other identities Dennis Eichhorn Department of Mathematics University of California Irvine Irvine CA 92697-3875 deichhor@ James Mc Laughlin Mathematics Department West Chester University West Chester PA 19383 jmclaughl@ Andrew V. Sills Department of Mathematical Sciences 203 Georgia Avenue Room 3008 Georgia Southern University Statesboro GA 30460-8093 ASills@ Submitted Sep 10 2010 Accepted Feb 24 2011 Published Mar 11 2011 Mathematics Subject Classifications 11P84 11P81 Abstract We give hybrid proofs of the q-binomial theorem and other identities. The proofs are hybrid in the sense that we use partition arguments to prove a restricted version of the theorem and then use analytic methods in the form of the Identity Theorem to prove the full version. We prove three somewhat unusual summation formulae and use these to give hybrid proofs of a number of identities due to Ramanujan. Finally we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities. 1 Introduction The proof of a q-series identity whether a series-to-series identity such as the second iterate of Heine s transformation see below a basic hypergeometric summation formula such as the q-Binomial Theorem see or one of the Rogers-Ramanujan identities see S14 below generally falls into one of two broad camps. In the one camp there are a variety of analytic methods. These include but are certainly not limited to elementary q-series manipulations as in the proof of the Bailey-Daum summation formula on page 18 of 15 the use of difference operators as in THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P60 1 Gasper and Rahman s derivation of a bibasic summation formula 14 the use of Bailey pairs and WP-Bailey pairs see for example 7 29 31 determinant methods for example 17 26 constant term methods such as in 4 Chap. 4 polynomial finitiza-tion .

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