Note on extended hypergeometric function


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

Biografía del autor

Ranjan Kumar Jana, Sardar Vallabhbhai National Institute of Technology.
Department of Applied Mathematics & Humanities.
Bhumika Maheshwari, Sardar Vallabhbhai National Institute of Technology.
Department of Applied Mathematics & Humanities.
Ajay Kumar Shukla, Sardar Vallabhbhai National Institute of Technology.
Department of Applied Mathematics & Humanities.


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

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.

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.

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.

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.

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.

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.

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:

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:

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

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.

E. Rainville, Special Functions, New York, NY: The Macmillan Company, 1960.

M. Saigo and N. Maeda, “More generalization of fractional calculus”, in Transform methods & special functions, Varna '96: second international workshop : proceedings, 1998, pp. 386–400.

H. Srivastava and P. Karlsson, Multiple gaussian hypergeometric series. Chichester: Horwood, 1985.

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:

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

Cómo citar
R. Jana, B. Maheshwari, y A. Shukla, Note on extended hypergeometric function, Proyecciones (Antofagasta, En línea), vol. 38, n.º 3, pp. 585-595, ago. 2019.