Note on extended hypergeometric function

Jana, Ranjan Kumar; Maheshwari, Bhumika; Shukla, Ajay Kumar; Jana, Ranjan Kumar; Maheshwari, Bhumika; Shukla, Ajay Kumar

doi:10.22199/issn.0717-6279-2019-03-0037

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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

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

http://dx.doi.org/10.22199/issn.0717-6279-2019-03-0037

Articles

Note on extended hypergeometric function

Ranjan Kumar Jana¹
http://orcid.org/0000-0001-5017-4189

Bhumika Maheshwari²

Ajay Kumar Shukla³
http://orcid.org/0000-0002-2713-2017

^¹Sardar Vallabhbhai National Institute of Technology (SVNIT), Department of Applied Mathematics & Humanities, Surat, Gujarat, India. e-mail : rkjana2003@yahoo.com

^²Sardar Vallabhbhai National Institute of Technology (SVNIT), Department of Applied Mathematics & Humanities, Surat, Gujarat, India. e-mail : bhumi0512@gmail.com

^³Sardar Vallabhbhai National Institute of Technology (SVNIT), Department of Applied Mathematics & Humanities, Surat, Gujarat, India. e-mail : ajayshukla2@rediffmail.com

Abstract

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.

Acknowledgement

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.

References

[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: http://bit.ly/2ZHWi1H [ Links ]

[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: http://bit.ly/2OPTb71 [ Links ]

[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: http://bit.ly/2OKSD1V [ Links ]

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

Received: May 2018; Accepted: July 2018

Article copyright: ©2019 Ranjan Kumar Jana, Bhumika Maheshwari and Ajay Kumar Shukla. This is an open access article distributed under the terms of the Creative Commons Licence, which permits unrestricted use and distribution provided the original author and source are credited

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