Vol. 10, Issue 4, Part B (2024)

Abstract
This paper presents a comprehensive analysis of basic hypergeometric functions and their generalizations, with particular emphasis on their relationship to q-difference calculus. We explore the fundamental properties of q-shifted factorials, basic hypergeometric series, and their applications in combinatorics, number theory, and mathematical physics. The study examines the convergence properties, transformation formulas, and connection formulas for these functions, while highlighting their role in modern q-calculus. Special attention is given to the Heine transformation, Bailey's transformation, Watson's transformation, and Jackson's q-integral representation. We also investigate the relationship between q-difference equations and basic hypergeometric functions, including the development of q-analogs of classical differential equations. The paper concludes with extensive applications to partition theory, quantum groups, orthogonal polynomials, and recent developments in elliptic hypergeometric functions. Additionally, we present computational algorithms and numerical methods for evaluating these functions.