Abstract
The purpose of this research is to provide a systematic review of a new type of extended beta function and hypergeometric function using a confluent hypergeometric function, as well as to examine various belongings and formulas of the new type of extended beta function, such as integral representations, derivative formulas, transformation formulas, and summation formulas. In addition, we also investigate extended Riemann–Liouville (R-L) fractional integral operator with associated properties. Furthermore, we develop new beta distribution and present graphically the relation between moment generating function and ℓ .
Highlights
Special functions, often denoted by series, emerge as a result of solving various problems in classical physics
Often denoted by series, emerge as a result of solving various problems in classical physics. These problems demand the flow of electromagnetic, acoustic, or thermal energy. It has developed into a vital resource for the particular and specialized roles played by scientists and engineers these days. ese characteristics are critical in a variety of domains, including physical science, mathematics, and engineering. e Gaussian hypergeometric function 2F1 is among most important special functions. e hypergeometric function 2F1 and its numerous generalizations have been investigated by many researchers who have provided a notably large number of their formula properties
Many researchers have established the generalizations of the classical gamma and beta functions and provided a number of intriguing and useful properties for the extended functions
Summary
Often denoted by series, emerge as a result of solving various problems in classical physics. These problems demand the flow of electromagnetic, acoustic, or thermal energy. Many researchers have established the generalizations of the classical gamma and beta functions and provided a number of intriguing and useful properties for the extended functions (see, e.g., [8,9,10,11,12,13,14,15,16]). Chaudhary et al [3] used the extended beta function (2) to present the following extension of the hypergeometric function: Fρ(a, b; c;. We develop a novel property-rich extension of the fractional derivative Riemann–Liouville operator
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