History, development and application of basic Hypergeometric Series

Abstract

The basic hypergeometric series, a powerful tool in the field of mathematical analysis, traces its origins to the study of special functions and q-series. Introduced by mathematicians in the early 20th century, it has since evolved to play a crucial role in various areas of mathematics, physics, and engineering. The basic hypergeometric series generalizes the classical hypergeometric series by incorporating a parameter, q, which leads to new and more flexible representations of mathematical functions. This series has found applications in combinatorics, number theory, quantum mechanics, and the study of orthogonal polynomials, among other fields. In this study, we explore the historical development of the basic hypergeometric series, beginning with its early formulations and progressing through its modern applications. Methodologically, we examine key contributions from mathematicians such as Ramanujan and Askey, who expanded the series’ scope and utility. The analysis includes an exploration of its convergence properties, its relationship with other special functions, and the application of the series in various scientific contexts. The expected outcomes of this study include a comprehensive understanding of the evolution of the basic hypergeometric series and a clear presentation of its modern-day applications. By highlighting its relevance in both theoretical and applied contexts, this work aims to demonstrate the versatility and importance of the basic hypergeometric series in contemporary mathematics and science. Ultimately, this research aims to provide a foundation for further exploration and innovation in the application of q-series.