The k-Confluent Hypergeometric Function and its properties in Bicomplex Numbers

In this paper, we examine a specialized form of the bicomplex hypergeometric function, known as the $k$-bicomplex confluent hypergeometric function (CHF). We introduce a detailed analysis of its properties, focusing on its formulation with bicomplex parameters, convergence conditions, and derivative and integral representations. By exploring the $k$-confluent case, we highlight unique theoretical insights and practical applications, particularly within the framework of bicomplex $k$-Riemann-Liouville (R-L) Fractional calculus. Our findings expand the current understanding of bicomplex functions in applied sciences and mathematical analysis, laying a foundation for further exploration in specialized functions and fractional operators within the bicomplex domain.In this paper, we examine a specialized form of the bicomplex hypergeometric function, known as the $k$-bicomplex confluent hypergeometric function (CHF). We introduce a detailed analysis of its properties, focusing on its formulation with bicomplex parameters, convergence conditions, and derivative and integral representations. By exploring the $k$-confluent case, we highlight unique theoretical insights and practical applications, particularly within the framework of bicomplex $k$-Riemann-Liouville (R-L) Fractional calculus. Our findings expand the current understanding of bicomplex functions in applied sciences and mathematical analysis, laying a foundation for further exploration in specialized functions and fractional operators within the bicomplex domain.