Rad HAZU, Matematičke znanosti, Vol. 30 (2026), 125-140. \( \)

GENERALIZED HORN FUNCTION \(H_{4,p,q,\nu}^\lambda\) AND RELATED BOUNDING INEQUALITIES WITH APPLICATIONS TO STATISTICS

Rakesh K. Parmar, Tibor K. Pogány and S. Pirivina

Department of Mathematics, Ramanujan School of Mathematical Sciences, Pondicherry University, Puducherry-605014, India
e-mail:rakeshparmar27@gmail.com, rakeshparmar@pondiuni.ac.in

Institute of Applied Mathematics, Óbuda University, 1034 Budapest, Hungary
Faculty of Maritime Studies, University of Rijeka, 51 000 Rijeka, Croatia
e-mail:pogany.tibor@nik.uni-obuda.hu, tibor.poganj@uniri.hr

Department of Mathematics, Ramanujan School of Mathematical Sciences, Pondicherry University, Puducherry-605014, India
e-mail:pirivina1610@pondiuni.ac.in

Abstract.   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 \(J_\nu(z)\) and modified Bessel functions \(I_\nu(z)\) for the generalized Horn double hypergeometric function \(H_{4,p,q,\nu}^\lambda\). Several functional upper bounds for the \(H_{4,p,q,\nu}^\lambda\) including the extended Gaussian hypergeometric \(F_{p,q,\nu}^\lambda\), the extended Kummer's confluent hypergeometric \(\Phi_{p,q,\nu}^\lambda\) are obtained by using functional bounds for extended Euler's Beta function \({\rm B}_{p,q,\nu}^\lambda(x,y)\). 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.

2020 Mathematics Subject Classification.   Primary: 26D15, 33C20, 33C65, 33C70; Secondary: 26D20; 33C70, 60E05

Key words and phrases.   Extended Beta function, Extended hypergeometric function, Extended confluent hypergeometric function, Horn double hypergeometric function \(H_4\), Bessel and modified Bessel functions, functional bounding inequalities, probability distribution, Turán inequalities

Full text (PDF) (free access)

https://doi.org/10.21857/y6zolb7wwm

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