Theta Function Identities of Level 16 and Their Applications to Colored Partitions

Theta function identities play an important role in the theory of modular forms and partitions, with Ramanujan’s q-series providing a fundamental analytical framework. This work derives several new identities of theta functions at level 16 by employing classical transformation techniques from the theory of q-series initiated by Ramanujan. Four main theorems are proved, yielding explicit algebraic relations involving the functions ϕ(q), ψ(q), and their associated transforms. These identities are applied to partition theory through the introduction of colored partition functions σi(m) defined by arithmetic conditions modulo 16. The resulting formulas provide direct combinatorial interpretations of the analytic identities and highlight connections between theta functions, modular forms, and partition functions. The results offer a unified analytic combinatorial framework at level 16 and contribute to the study of modular congruences and related arithmetic structures.