On some subclasses of hypergeometric functions with Djrbashian Cauchy type kernel


Joel Esteban Restrepo - Institute of Mathematics, University of Antioquia, Cl. 53 - 108, Medellin, Colombia.
Armen Jerbashian - Institute of Mathematics, University of Antioquia, Cl. 53 - 108, Medellin, Colombia.
Praveen Agarwal - Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India.


In this paper, some new integral representations are proved for several weighted hypergeometric functions introduced recently in [J. E. Restrepo, A. Kılıc¸man, P. Agarwal, O. Altun, Adv. Difference Equ., 2017 (2017), 11 pages]. Besides, some new subclasses of weighted hypergeometric functions containing the Djrbashian Cauchy type kernel are introduced. The series representing the considered hypergeometric functions are convergent out of some sets of zero !-capacity, and these hypergeometric functions have finite boundary values everywhere on \(|z|=1\), out of zero \(\omega\)-capacity sets.



[1] P. Agarwal, J.-S. Choi, K. B. Kachhia, J. C. Prajapati, H. Zhou,/ Some integral transforms and fractional integral formulas for the extended hypergeometric functions,/ Commun. Korean Math. Soc.,/ 31 (2016), 591–601.
[2] W. N. Bailey,/ Generalized hypergeometric series,/ Cambridge Tracts in Mathematics and Mathematical Physics, Stechert-Hafner, Inc., New York,/ (1964).
[3] M. A. Chaudhry, A. Qadir, M. Rafique, S. M. Zubair,/ Extension of Euler’s beta function,/ J. Comput. Appl. Math.,/ 78 (1997), 19–32.
[4] M. A. Chaudhry, A. Qadir, H. M. Srivastava, R. B. Paris,/ Extended hypergeometric and confluent hypergeometric functions,/ Appl. Math. Comput.,/ 159 (2004), 589–602.
[5] A. M. Dzhrbashyan,/ An extension of the factorization theory of M. M. Dzhrbashyan,/ (Russian); translated from Izv. Nats. Akad. Nauk Armenii Mat., 30 (1995), 47–75, J. Contemp. Math. Anal.,/ 30 (1995), 39–61.
[6] M. M. Dzhrbashyan, V. S. Zakharyan,/ Klassy i granichnye svoĭstva funktsiĭ, meromorfnykh v kruge,/ (Russian) [[Classes and boundary properties of functions that are meromorphic in the disk]] Fizmatlit “Nauka”, Moscow,/ (1993).
[7] M. M. Džrbašjan,/ Theory of factorization and boundary properties of functions meromorphic in the disk,/ Proceedings of the International Congress of Mathematician, Vancouver, B. C., (1974), Canad. Math. Congress, Montreal, Que.,/ 2 (1975), 197–202.
[8] O. Frostman,/ Potentiel d’équilibre et capacité des ensembles avec quelques applications a la théorie des fonctions,/ (French) Madd. Lunds. Univ. Mat. Sem.,/ 3 (1935), 1–11.
[9] O. Frostman,/ Sur les produits de Blaschke,/ Fysiogr. Säldsk. Lund, föhr.,/ 12 (1939), 1–14.
[10] O. Frostman,/ Sur les produits de Blaschke,/ (French) Kungl. Fysiografiska S¨allskapets i Lund Förhandlingar [Proc. Roy. Physiog. Soc. Lund],/ 12 (1942), 169–182.
[11] ˙I. O. Kiymaz, A. Çetinkaya, P. Agarwal,/ An extension of Caputo fractional derivative operator and its applications,/ J. Nonlinear Sci. Appl.,/ 9 (2016), 3611–3621.
[12] L. K. B. Kuroda, A. V. Gomes, R. Tavoni, P. F. de Arruda Mancera, N. Varalta, R. de Figueiredo Camargo,/ Unexpected behavior of Caputo fractional derivative,/ Comput. Appl. Math.,/ (2016), 1–11.
[13] E. Özergin,/ Some properties of hypergeometric functions,/ Ph.D. Thesis, Eastern Mediterranean University, North Cyprus, Turkey,/ (2011).
[14] J. E. Restrepo, A. Kılıçman, P. Agarwal, O. Altun,/ Weighted hypergeometric functions and fractional derivative,/ Adv. Difference Equ.,/ 2017 (2017), 11 pages.
[15] V. E. Tarasov,/ Some identities with generalized hypergeometric functions,/ Appl. Math. Inf. Sci.,/ 10 (2016), 1729–1734.
[16] X.-J. Yang, D. Baleanu, H. M. Srivastava,/ Local fractional integral transforms and their applications,/ Elsevier/Academic Press, Amsterdam,/ (2016).
[17] X.-J. Yang, H. M. Srivastava,/ An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives,/ Commun. Nonlinear Sci. Numer. Simul.,/ 29 (2015), 499–504.