A New Generalized Beta Function Associated with Statistical Distribution and Fractional Kinetic Equation

Several authors have extensively investigated beta function, hypergeometric function, and confluent hypergeometric function, their extensions, and generalizations due to their several applications in many areas of engineering, probability theory, and science. The main purpose of this paper is to present a new generalization of the extended beta function, hypergeometric function, and confluent hypergeometric function with the help of the m-parameter Mittag-Leffler function, as well as examine some important properties like integral representations, differential formulas, and summation formulas. We also examine the generalized Caputo fractional derivative operator with the help of the m-parameter Mittag-Leffler function and associated properties using the generalized beta function. We define a new beta distribution involving the new generalized beta function. The mean, variance, coefficient of variance, moment generating function, characteristic function, and cumulative distribution are derived. Further, we derive the solution of a fractional kinetic equation involving generalized hypergeometric functions.