Generalized Horn function \ (H₄, ₏, ₐ, ^\) 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 \ (H₄x, y\). We then establish integral representations of the Euler and Laplace type including some other representations involving Bessel \ (J_ (z) \) and modified Bessel functions \ (I_ (z) \) for the generalized Horn double hypergeometric function \ (H₄, ₏, ₐ, ^\). Several functional upper bounds for the \ (H₄, ₏, ₐ, ^\) including the extended Gaussian hypergeometric \ (F₏, ₐ, ^\), the extended Kummer's confluent hypergeometric \ (₏, ₐ, ^\) are obtained by using functional bounds for extended Euler's Beta function \ (B₏, ₐ, ^ (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.