The Incomplete First Appell Hypergeometric Functions

Hypergeometric functions belong to the field of special function and play an im-portant role not only in combinatorics, number theory and mathematical equations, but also in physics, control engineering, communications engineering, and other areas.Srivastava et al. established the incomplete Gamma functions by Gamma function and presented its application in physics and statistics in 2012. Cetinkaya constructed the incomplete second Appell function by incomplete Gamma function and studied its prop-erties in 2013. Inspired by Srivastava et al.’s work on the incomplete Pochhammer sym-bols and their applications to hypergeometric and related functions, and then Cetinkaya’s researches on the incomplete second Appell hypergeometric functions, the authors define the incomplete first Appell hypergeometric functions and several properties of this func-tion are established. In this paper, we present three chapters to exposit our research of the incomplete first Appell hypergeometric function.In the first chapter, we introduce the basic concepts, properties, historical back-ground and current situation of hypergeometric functions.In the second chapter, we introduce the properties of Appell function and the in-complete second Appell function that was exposited.In the third chapter, by introducing the incomplete Beta function, the authors con-struct the new incomplete Appell function and research the properties of the incomplete first Appell functions by the method of parametric replacement and series rearrangement, and then obtain several new differential properties, integral transformations and recursion formula.In the four chapter, conclude the main contents of this paper and look forward to further research.