TAILIEUCHUNG - Báo cáo toán học: " Polynomial Generalizations of two-variable Ramanujan type 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: Polynomial Generalizations of two-variable Ramanujan type identities. | Polynomial Generalizations of two-variable Ramanujan type identities James Mc Laughlin Department of Mathematics 124 Anderson Hall West Chester University West Chester PA 19383 USA jmclaughlin@ Andrew V. Sills Department of Mathematical Sciences 203 Georgia Avenue Georgia Southern University Statesboro GA 30460 USA asills@ Submitted Jun 26 2010 Accepted Jun 14 2011 Published Jun 28 2011 Mathematics Subject Classifications 11B65 05A10 Dedicated to Doron Zeilberger on the occasion of his sixtieth birthday. The progress of mathematics can be viewed as progress from the infinite to the finite. Gian-Carlo Rota 1983 Abstract We provide finite analogs of a pair of two-variable q-series identities from Ramanujan s lost notebook and a companion identity. 1 Introduction At the top of a page in the lost notebook 14 p. 33 cf. 6 p. 99 Entry Ramanujan recorded an identity equivalent to the following j 0 q2j2 zq q2 j q z q2 j q2 q2 2j zq3 q3 z q6 q q2 q2u. where we are employing the standard notation for rising q-factorials A qU. 1 A 1 Aq 1 Aq2 and A q n A qK Aqn q THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P15 1 and A1 A2 Ar q w A1 qW -A2 q TO Ar q W- In a recent paper 11 we found a partner to that Ramanujan appears to have missed j 0 qj j 1 z q j q z q j i q q 2j 1 zq2 q z q3 q3 x q qA. Later on the same page of the lost notebook Ramanujan recorded 6 p. qj2 zq q2 j q z q2 j zq2 q2 z q4 q4 U-q q2 x j 0 q q2 j q4 q4 j q2 q2O. 103 Entry For further discussion of these three identities see 10 . Remark. Out of respect for Doron s ultra-finitist philosophy we deliberately refrain from stating conditions on q and z which imply analytic convergence of the infinite series and products in - . The preceding identities stand out among identities of Rogers-Ramanujan type because they are two-variable series-product identities. While Rogers-Ramanujan type identities admit two-variable generalizations most lose the .

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