THETA FUNCTION IDENTITIES AND RAMANUJAN’S CONGRUENCES ON THE PARTITION FUNCTION by WENCHANG CHU†

A q-difference equation on eight shifted factorials of infinite order will be established. As conse-quences, 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 ðq; qÞ101 and there-fore a new proof of the Ramanujan congruence on the partition function modulo 11. For two indeterminates q and z with jqj, 1; the q-shifted factorial of infinite order and the theta function are defined respectively by ðz; qÞ