Generalized Horn function \(H4,p,q,ν^λ\) 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 \(J_ν(z)\) and modified Bessel functions \(I_ν(z)\) for the generalized Horn double hypergeometric function \(H4,p,q,ν^λ\). Several functional upper bounds for the \(H4,p,q,ν^λ\) including the extended Gaussian hypergeometric \(Fp,q,ν^λ\), the extended Kummer's confluent hypergeometric \(Φp,q,ν^λ\) are obtained by using functional bounds for extended Euler's Beta function \({ B}p,q,ν^λ(x,y)\). 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.