| By means of classical analytic method and combinatorial computational technique, such as functional equations, Liouville theorem, series rearrangements, this dissertation investigates the product identities on theta functions. Besides giving new proofs of known identities, the author establishes several interesting product identities, and discusses briefly applications to combinatorial computations as well as analytic theory of numbers. The content is summarized as follows:1. By means of two different methods, the author derives a general equation for expanding the product of two Jacobi's triple products, which can be considered as the common generalization of the quintuple, sextuple and septuple product identities. As main consequences, the author explores briefly its applications to identities for certain products of theta functions (?)(q) andψ(q) and modular relations for the Gollnitz-Gordon functions.2. Inspired by Chan's double series representation formula of (q;q)∞10, the author constructs a new difference equation on quintuple products, which leads to clarification on the four symmetric differences that are equivalent in pairs. In addition, new proofs are presented for two symmetric difference identities, as well as Ramanujan's congruence modulo 11 on the partition function.3. Liouville's Theorem on entire functions is proposed as a proving method for theta function identities. Several identities are exemplified, such as the quintuple and septuple product identities; four symmetric difference identities related to the Ramanujan's congruence modulo 11 on the partition function; as well as two theta function identities on the Rogers-Ramanujan functions G(q) and H(q).4. By means of Jacobi's triple product identity and its linear combination, the author derives several interesting identities, including theta function formulae due to Baruah-Berndt [12, 13], identities of Rogers-Ramanujan functions [14, 75, 82] and modular equations due to Ramanujan [9, Thm 1.6.1].
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