A q-difference equation on eight shifted factorials of infinite order will be established. As consequences, we shall systematically explore triple products to give alternative proofs of the theta function identities due to Ewell [5, 6] and Berndt et al. [2] and briefly review their applications to the Ramanujan congruences on the partition function. In particular, a q-difference equation on quintuple products will be discovered which leads us to a new representation of $(q; q)^{10}_\infty$ and therefore a new proof of the Ramanujan congruence on the partition function modulo 11.
Theta Function Identities and Ramanujan's Congruences on Partition Function
CHU, Wenchang
2005-01-01
Abstract
A q-difference equation on eight shifted factorials of infinite order will be established. As consequences, we shall systematically explore triple products to give alternative proofs of the theta function identities due to Ewell [5, 6] and Berndt et al. [2] and briefly review their applications to the Ramanujan congruences on the partition function. In particular, a q-difference equation on quintuple products will be discovered which leads us to a new representation of $(q; q)^{10}_\infty$ and therefore a new proof of the Ramanujan congruence on the partition function modulo 11.File in questo prodotto:
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