SciELO - Scientific Electronic Library Online

vol.38 issue3A transmuted version of the generalized half-normal distributionPebbling on zig-zag chain graph of n odd cycles author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand




Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.38 no.3 Antofagasta Aug. 2019 


Note on extended hypergeometric function

Ranjan Kumar Jana1

Bhumika Maheshwari2 

Ajay Kumar Shukla3

1Sardar Vallabhbhai National Institute of Technology (SVNIT), Department of Applied Mathematics & Humanities, Surat, Gujarat, India. e-mail :

2Sardar Vallabhbhai National Institute of Technology (SVNIT), Department of Applied Mathematics & Humanities, Surat, Gujarat, India. e-mail :

3Sardar Vallabhbhai National Institute of Technology (SVNIT), Department of Applied Mathematics & Humanities, Surat, Gujarat, India. e-mail :


In this paper, we present an extension of the classical hypergeometric functions using extended gamma function due to Jumarie and obtained some properties.

Keywords: Gamma function; Pochhammer symbols; Hypergeometric functions; Integral transforms; Fractional calculus.

Mathematics Subject Classification (2010):  33B15; 33C05; 33C20; 44A05; 26A33

Texto completo disponible sólo en PDF.

Full text available only in PDF format.


This work was partially supported by SERB, Govt. of India, Research Project No. EMR/2016/000351, sanctioned to the author RKJ. The authors are grateful to the anonymous referee(s) for their valuable suggestions, which resulted in the improvement of the paper.


[1] E. Arthur, W. Magnus, F. Oberhettinger, F. Tricomi, and H. Bateman,Higher transcendental functions, vol. 1. New York, NY: McGraw-Hill, 1953. [ Links ]

[2] R. Gorenflo, A. Kilbas, F. Mainardi, and S. Rogosin,Mittag-Leffler Functions, Related Topics and Applications. Berlin, Heidelberg: Springer, 2014, doi: 10.1007/978-3-662-43930-2. [ Links ]

[3] G. Jumarie, “On the representation of fractional Brownian motion as an integral with respect to (dt)a”,Applied Mathematics Letters, vol. 18, no. 7, pp. 739-748, Jul. 2005, doi: 10.1016/j.aml.2004.05.014. [ Links ]

[4] G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results”,Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367-1376, May 2006, doi: 10.1016/j.camwa.2006.02.001. [ Links ]

[5] G. Jumarie , “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions”,Applied Mathematics Letters , vol. 22, no. 3, pp. 378-385, Mar. 2009, doi: 10.1016/j.aml.2008.06.003. [ Links ]

[6] G. Jumarie , “Laplace’s transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative”,Applied Mathematics Letters , vol. 22, no. 11, pp. 1659-1664, Nov. 2009, doi: 10.1016/j.aml.2009.05.011. [ Links ]

[7] G. Jumarie , “Cauchy’s integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order”,Applied Mathematics Letters , vol. 23, no. 12, pp. 1444-1450, Dec. 2010, doi: 10.1016/j.aml.2010.08.001. [ Links ]

[8] G. Jumarie , “Fractional Euler’s integral of first and second kinds. Application to fractional Hermite’s polynomials and to probability density of fractional order”, Journal of Applied Mathematics and Informatics, vol. 28, no. 1-2, pp. 257-273, 2010. [On line]. Available: ]

[9] G. Jumarie , “The Leibniz rule for fractional derivatives holds with non-differentiable functions”, Mathematics and Statistics, vol. 1, no. 2, pp. 50-52, 2013. [On line]. Available: ]

[10] G. Jumarie,Fractional differential calculus for non-differentiable functions: mechanics, geometry, stochastics, information theory. Saarbrücken: LAP LAMBERT Academic Publishing, 2013. [ Links ]

[11] G. Jumarie , “On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling”, Central European Journal of Physics, vol. 11, no. 6, pp. 617-633, Oct. 2014, doi: 10.2478/s11534-013-0256-7. [ Links ]

[12] E. Rainville, Special Functions, New York, NY : The Macmillan Company, 1960. [ Links ]

[13] M. Saigo and N. Maeda, “More generalization of fractional calculus”, inTransform methods & special functions, Varna '96: second international workshop : proceedings, 1998, pp. 386-400. [ Links ]

[14] H. Srivastava and P. Karlsson,Multiple gaussian hypergeometric series. Chichester: Horwood, 1985. [ Links ]

[15] N. Virchenko, “On the generalized confluent hypergeometric function and its applications”, Fractional Calculus and Applied Analysis, vol. 9, no. 2, pp. 101-108, 2006. [On line]. Available: ]

[16] N. Virchenko, “On some generalizations of classical integral transforms”. Mathematica. Balkanica, vol. 26, no. 1-2, 2012. [On line]. Available: ]

Received: May 2018; Accepted: July 2018

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License