A New Extension of Beta Function by Associated with Appell Function

The main objective of this paper is to introduce a further extension of extended beta function by considering Appell functions. We investigate various properties of this newly defined beta function such as integral representations, summation formulas, generalized Gauss and confluent hypergeometric functions and differentiation formulas for the extended hypergeometric and confluent hypergeometric functions. Further, some new relations for various forms of extended beta functions are obtained as special cases of the main results.