Abstract
In 1988, N.J. Fine published a monograph entitled Basic Hypergeometric Series and Applications in which he proved a number of results concerning the series F(a,b;t:q). In this paper, we present a new combinatorial interpretation for the series F(a,b;t:q) and use Fine’s work as a guide for proving the Rogers–Fine identity and many of its properties in this setting.
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References
Andrews, G.E.: The Theory of Partitions. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1998). Reprint of the 1976 original
Fine, N.J.: Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, vol. 27. American Mathematical Society, Providence (1988). With a foreword by George E. Andrews
Rogers, L.J.: On two theorems of combinatory analysis and some allied identities. Proc. Lond. Math. Soc. 16(2), 315–336 (1917)
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Little, D.P. Some Fine combinatorics. Ramanujan J 23, 341–353 (2010). https://doi.org/10.1007/s11139-009-9195-8
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DOI: https://doi.org/10.1007/s11139-009-9195-8