Generalized Horn function Hλ 4,p,q,ν and related bounding inequalities with applications to statistics

Motivated by recent unified version of the Euler's Beta integral form with a MacDonald function in the integrand, we generalize the Horn double hypergeometric function H 4 [x, y].We then establish integral representations of the Euler and Laplace type including some other representations involving Bessel Jν (z) and modified Bessel functions Iν (z) for the generalized Horn double hypergeometric function H λ 4,p,q,ν .Several functional upper bounds for the H λ 4,p,q,ν including the extended Gaussian hypergeometric F λ p,q,ν , the extended Kummer's confluent hypergeometric Φ λ p,q,ν are obtained by using functional bounds for extended Euler's Beta function B λ p,q,ν (x, y).Various other bounding inequalities are obtained via Luke's, von Lommel's, Minakshisundaram and Szász and Olenko bounds.As an application, we define a Horn hypergeometric probability distribution to obtain certain statistical interference.