Extended Exton’s Triple and Horn’s Double Hypergeometric Functions and Associated Bounding Inequalities

This paper introduces extensions H4,p and X8,p of Horn’s double hypergeometric function H4 Exton’s triple X8, taking into account recent Euler’s beta function, confluent function. Among the numerous extended functions, primary rationale for choosing X8 is their comparable extension type. Next, we present various integral representations Euler Laplace types, Mellin inverse transforms, Laguerre polynomial representations, transformation formulae, a recurrence relation functions. In particular, provide generating several bounding inequalities X8,p. We explore utilization within probability distribution. Most special such as generalized Beta p-extended integral, exhibit natural symmetry.