Two-Variable Hypergeometric Functions: Theory, Properties, and Applications (Leadership, Education, Personality: An Interdisciplinary Journal, ISSN: 2524-6178, 2025)
Two-variable hypergeometric functions represent a significant extension of classical hypergeometric functions, incorporating the complexity of multiple independent variables. These functions emerge in various mathematical and physical contexts, offering powerful tools for solving problems in areas such as differential equations, mathematical physics, and algebraic geometry. This paper provides a comprehensive exploration of the theory, properties, and applications of two-variable hypergeometric functions. We begin by reviewing their foundational definitions, integral representations, and key properties, including convergence, asymptotic behavior, and symmetries. The paper also discusses important special cases and generalizations, as well as recurrence relations and differential equations associated with these functions. In addition to their theoretical significance, we highlight several practical applications across diverse fields, including quantum mechanics, statistical mechanics, signal processing, and algebraic geometry. Finally, we examine recent advancements and open problems in the study of two-variable hypergeometric functions, identifying promising avenues for future research. This work aims to offer a unified framework for understanding the rich structure and broad applicability of two-variable hypergeometric functions, making them an indispensable tool in both pure and applied mathematics.