Generalizations of Schur's partition theorem

Abstract

Schur's partition theorem states that the number of partitions ofn into distinct parts ≡ 1,2 (mod 3) is equal to the number of partitions ofn into parts with minimal difference 3 and no consecutive multiples of 3. A three-parameter generalization of Gleissberg's refinement of Schur's theorem is obtained by showing that\(\Pi _{m = 1}^\infty (1 + aq^m )(1 + bq^m )\) is equal to the numerator of a certain continued fraction. Two proofs are presented, one completely combinatorial, and one using generating functions.