Exploring the Applications of Bicomplex Confluent Hypergeometric Functions in Fractional Calculus and Statistical Distributions

This paper introduces a special version of hypergeometric functions constructed within the bicomplex system, referred to as the bicomplex confluent hypergeometric functions (BCHFs). We highlight important properties, concentrating on their structure with bicomplex numbers, their conditions for convergence, and their differential representation. We showcase foundational theories and applied practices, especially in the context of bicomplex Riemann–Liouville (R–L) fractional calculus. Additionally, we explain the framework of BCHF in probability distribution theory and present critical findings, focusing on the statistical properties of a bicomplex-valued probability distribution. We provide a numerical analysis of the BCHF and the probability density function in bicomplex, including plot representations, to clarify the behavior and structure of the bicomplex components. This research lays the groundwork for future advancements in specialized functions, fractional operators, probability distribution theory, and numerical analysis within the bicomplex operator.