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∞(qt; qt)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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