Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities

Stembridge, John R.

doi:10.1090/S0002-9947-1990-0986702-5

Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities
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by John R. Stembridge

Trans. Amer. Math. Soc. 319 (1990), 469-498

DOI: https://doi.org/10.1090/S0002-9947-1990-0986702-5

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Abstract:

We apply the theory of Hall-Littlewood functions to prove several multiple basic hypergeometric series identities, including some previously known generalizations of the Rogers-Ramanujan identities due to G. E. Andrews and D. M. Bressoud. The techniques involve the adaptation of a method due to I. G. Macdonald for calculating partial fraction expansions of certain types of symmetric formal power series. Macdonald originally used this method to prove a pair of generating function identities for plane partitions conjectured by MacMahon and Bender-Knuth. We show that this method can also be used to prove another pair of plane partition identities recently obtained by R. A. Proctor.

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