S-generalized Mittag-Leffler Function and its Certain Properties

Year 2019, Volume: 7 Issue: 2, 139 - 148, 15.10.2019

Praveen Agarwal Aysegul Cetinkaya Shilpi Jain İ. Onur Kıymaz

https://doi.org/10.36753/mathenot.578638

Cited By: 2

Abstract

In 2014, S-generalized beta function which consist of
seven parameters, defined and studied by Srivastava et al. [H. M.
Srivastava, P. Agarwal and S. Jain, Generating functions for the
generalized Gauss hypergeometric functions, Appl. Math. Comput., 247 (2014), pp. 348-352]. In this paper, by using S-generalized
beta function, we introduce a new generalization of Mittag-Leffler
function. This new generalization of Mittag-Leffler function is consist of eleven parameters. We also investigate some of its certain
properties such as integral representations, recurrence formulas and
derivative formulas by using classical and fractional derivatives.
Furthermore, we determine its Mellin, beta and Laplace integral
transforms.  

Keywords

Mittag-Leffler function, generalized beta function, fractional derivative

Supporting Institution

Department of Science \& Technology(DST), India and Science \& Engineering Research Board (SERB), India

Project Number

INT/RUS/RFBR/P-308 and TAR/2018/000001

Thanks

The research was supported by the Department of Science \& Technology(DST), India (No:INT/RUS/RFBR/P-308) and Science \& Engineering Research Board (SERB), India (No:TAR/2018/000001).

References

• 1. R. P. Agarwal and P. Agarwal, Extended Caputo fractional derivativeoperator, Adv. Stud. Contemp. Math., 25:3 (2015), pp. 301-316.2. P. Agarwal, M. Chand and S. Jain, Certain integrals involving generalized Mittag-Leffler functions, Proc. Nat. Acad. Sci. India Sect.A, 85:3 (2015), pp. 359-371.3. P. Agarwal, J. Choi, S. Jain and M. M. Rashidi, Certain integrals associated with generalized Mittag-Leffler function, Commun. KoreanMath. Soc, 32:1 (2017), pp. 29-38.4. P. Agarwal, J. Choi and R. B. Paris, Extended Riemann-Liouvillefractional derivative operator and its applications, J. Nonlin. Sci.Appl., 8:5, (2015), pp. 451-466.5. P. Agarwal and J. J. Nieto, Some fractional integral formulas forthe Mittag-Leffler type function with four parameters, Open Math.,13:1 (2015), pp. 537-546.6. P. Agarwal, S. V. Rogosin and J. J. Trujillo, Certain fractional integral operators and the generalized multi-index Mittag-Leffler functions, Proc. Indian Acad. Sci. Math. Sci., 125:3 (2015), pp. 291-306.7. P.L. Butzer and S. Jansche, A direct approach to the Mellin transform, J. Fourier Anal., 3 (1997), pp. 325-376.8. R. F. Camargo, E. Capelas de Oliveira and J. Vas, On the generalizedMittag-Leffler function and its application in a fractional telegraphequation, Math. Phys. Anal. Geom., 15:1 (2012), pp. 1-16.9. M. Kurulay and M.Bayram, Some properties of the Mittag-Lefflerfunctions and their relation with the Wright function, Adv. Differ. Equ., 2012:181 (2012), https://doi.org/10.1186/1687-1847-2012-181.10. M. A. Chaudhry, A. Qadir, M. Rafique and S. M. Zubair, Extensionof Euler’s beta function, J. Comput. Appl. Math., 78 (1997), pp.19-32.11. M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris,Extended hypergeometric and confluent hypergeometric functions,Appl. Math. Comput., 159 (2004), pp. 589-602.12. R. Gorenflo, A. Kilbas, F. Mainardi and S. Rogosin, Mittag-Lefflerfunctions: Related topics and applications, Springer, Berlin, 2010.13. R. Hilfer, Fractional time evolution, in: R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific PublishingCompany, Singapore, New Jersey, London and Hong Kong, 2000.14. R. Hilfer and H. Seybold, Computation of the generalized MittagLeffler function and its inverse in the complex plane, Integral Transform. Spec. Funct., 17 (2006), pp. 637-652.15. A. A. Kilbas and M. Saigo, On Mittag-Leffler type function, fractional calculus operators and solutions of integral equations, IntegralTransform. Spec. Funct., 4 (1996), pp. 355-370.16. A. A. Kilbas, M. Saigo and R. K. Saxena, Generalized MittagLeffler function and generalized fractional calculus operators, Integral Transform. Spec. Funct., 15 (2004), pp. 31-49.17. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, 2006.18. M.-J. Luo, G. V. Milovanovic and P. Agarwal, Some results on theextended beta and extended hypergeometric functions, Applied Mathematics and Computation, 248 (2015), pp. 631-651.19. E. Ozergin, M.A. ¨ Oarslan and A. Altın, ¨ Extension of gamma, betaand hypergeometric function, J. Comput. Appl. Math., 235 (2011),pp. 4601-4610.20. M. A. Ozarslan and B. Yılmaz, ¨ The extended Mittag-Lefflerfunction and its properties, J. Inequal. Appl., 2014:85 (2014),https://doi.org/10.1186/1029-242X-2014-85.21. R. K. Parmar, A new generalization of Gamma, Beta, hypergeometric and confluent hypergeometric functions, Matematiche (Catania),69 (2013), pp. 33-52.22. T. R. Prabhakar, A singular integral equation with ageneralizedMittag-Leffler function in the kernel, Yokohama Math.J., 19 (1971), pp. 7-15.23. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach,Yverdon et al. (1993).24. M. Sharma and R. Jain, A note on a generalized M-series as a specialfunction of fractional calculus, Fract. Calc. Appl. Anal., 12:4 (2009),pp. 449-452.25. I. N. Sneddon, The Use of Integral Transforms, Tata McGraw-Hill,New Delhi, 1979.26. M. R. Spiegel, Theory and Problems of Laplace Transforms,Schaums Outline Series, McGraw-Hill, New York, 1965.27. H. M. Srivastava, P. Agarwal and S. Jain, Generating functions forthe generalized Gauss hypergeometric functions, Appl. Math. Comput., 247 (2014), pp. 348-352.28. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric Series, Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley &Sons, Inc.], New York, 1985.29. H. M. Srivastava, R. Jain and M. K. Bansal, A study of the Sgeneralized Gauss hypergeometric function and its associated integraltransforms, Turkish J. Anal. Number Theory, 3 (2015), pp. 101-104.