Powers of Euler's Product and Related Identities |
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Authors: | Shaun Cooper Michael D Hirschhorn Richard Lewis |
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Institution: | (1) IIMS, Massey University, Albany Campus, Private Bag 102 904, North Shore Mail Centre, Auckland, New Zealand;(2) School of Mathematics, UNSW, Sydney, Australia, 2052;(3) SMS, University of Sussex, Brighton, BN1 9QH, UK |
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Abstract: | Ramanujan's partition congruences can be proved by first showing that the coefficients in the expansions of (q; q)
r
satisfy certain divisibility properties when r = 4, 6 and 10. We show that much more is true. For these and other values of r, the coefficients in the expansions of (q; q)
r
satisfy arithmetic relations, and these arithmetic relations imply the divisibility properties referred to above. We also obtain arithmetic relations for the coefficients in the expansions of (q; q)
r
(q
t; q
t)
s
, for t = 2, 3, 4 and various values of r and s. Our proofs are explicit and elementary, and make use of the Macdonald identities of ranks 1 and 2 (which include the Jacobi triple product, quintuple product and Winquist's identities). The paper concludes with a list of conjectures. |
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Keywords: | Ramanujan's partition congruences Macdonald identities Winquist's identity Euler's product |
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