New Mock Theta Function Identities via Fractional q-Calculus and Bilateral 2ψ2 Series

Mock theta functions, introduced by Ramanujan in his last letter to Hardy, play a significant role in q-series theory and have natural connections to fractional q-calculus. In this paper, we study bilateral hypergeometric series of the form ψ22= ∑n=−∞∞(a,b;q)n(c,d;q)nzn, where (a;q)n denotes the q-shifted factorial. Using Slater’s three-term transformation formula for bilateral ψ22 series, we derive new identities for Ramanujan’s mock theta functions of orders 2, 3, 6, and 8. These transformations reveal previously unknown relationships between different q-series representations and extend the classical theory of mock theta functions within the framework of q-special functions.