Analytical studies of the complex euler-beta integral functions generated by modified bessel-maitland type functions

In recent years, the continuous evolution of various branches of mathematics and other disciplines has led to the emergence of novel classes of special functions (SFs), their generalizations, and their extensions. The Bessel function is intricately connected to the solutions of certain differential equations, fostering research and innovation. The Bessel-maitland function (BM-function) is a notable generalized Bessel function. This paper proposes a new equation for the extended Euler-Beta function (EB-function) using the modified BM-function as its kernel, termed the Euler-Beta-Bessel-Maitland function (EBBM-function). Moreover, many features and formulas of the constructed EBBM-function are analyzed, including summation formulas, functional relationships, integral representation formulas, and derivative formulas. The statistical distribution related to the EBBM-function is also discussed. Furthermore, extended Kummer functions (K-functions) and Gauss hypergeometric functions (GH-functions) influenced by the EBBM-function are constructed, and their varied properties, including integral representations, derivative formulas, and generating functions, are investigated.