Fractional calculus formulas for Mathieu-type series and generalized Mittag-Leffler function

Volume 20, Issue 2, pp 122--130 http://dx.doi.org/10.22436/jmcs.020.02.05

Publication Date: October 29, 2019 Submission Date: July 18, 2019 Revision Date: September 05, 2019 Accteptance Date: September 17, 2019

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Authors

Owais Khan - Department of Applied Mathematics, Aligarh Muslim University, Aligarh-202002, India. Serkan Araci - Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, TR-27410, Gaziantep, Turkey. Mohd Saif - Department of Applied Mathematics, Aligarh Muslim University, Aligarh-202002, India.

Abstract

Fractional calculus is allowing integrals and derivatives of any positive order (the term `fractional' kept only for historical reasons), which can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power-law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this line, our main object to investigate image formulas of generalized fractional hypergeometric operators involving the product of Mathieu-type series and generalized Mittag-Leffler function. We also consider some interesting special cases of derived results by specializing suitable value of the parameters.

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Keywords

• Fractional calculus operators
• Mathieu-type series
• generalized Mittag-Leffler function
• Fox-Wright function

MSC

•  26A33
•  33C20
•  33B15

References

• [1] P. Agarwal, J. J. Nieto, Some fractional integral formulas for the Mittag-Leffler type function with four parameters, Open Math., 13 (2015), 537--546
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] S. Ahmed, On the generalized fractional integrals of the generalized Mittag-Leffler function, SpringerPlus, 3 (2014), 5 pages
  • View Article
  • Google Scholar


• [3] S. Araci, G. Rahman, A. Ghaffar, Azeema, K. S. Nisar, Fractional calculus of extended Mittag-Leffler function and its applications to statistical distribution, Mathematics, 7 (2019), 14 pages
  • View Article
  • Google Scholar


• [4] M. Arshad, J. S. Choi, S. Mubeen, K. S. Nisar, G. Rahman, A new extension of the Mittag-Leffler function, Commun. Korean Math. Soc., 33 (2018), 549--560
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] M. Bohner, G. Rahman, S. Mubeen, K. S. Nisar, A further extension of the extended Riemann-Liouville fractional derivative operator, Turkish J. Math., 42 (2018), 2631--2642
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] P. Cerone, C. T. Lenard, On integral forms of generalized Mathieu series, J. Inequal. Pure Appl. Math., 4 (2003), 1--11
  • View Article
  • Google Scholar


• [7] O. Emersleben, Über die Reihe $\sum^\infty_{k=1} k/{(k^2+c^2)^2}$, Math. Ann., 125 (1952), 165--171
  • View Article
  • MathSciNet
  • MATH


• [8] C. Fox, The $G$ and $H$ functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc., 98 (1961), 395--429
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] R. Gorenflo, F. Mainardi, Fractional calculus, integral and differential equations of fractional order, in: Fractals and Fractional Calculus in Continuum Mechanics, 1997 (1997), 223--279
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] R. Hilfer, Application of fractional Calculus in Physics, World Scientific Publishing Co., River Edge (2000)
  • View Article
  • MathSciNet
  • Google Scholar


• [11] C. M. Jiang, F. F. Zhang, T. X. Li, Synchronization and Antisynchronization of N-coupled Fractional-Order Complex Chaotic Systems with Ring Connection, Math. Methods Appl. Sci., 41 (2018), 2625--2638
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] M. Kamarujjama, O. Khan, Computation of new class of integrals involving generalized Galue type Struve function, J. Comput. Appl. Math., 351 (2019), 228--236
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] M. Kamarujjama, N. U. Khan, O. Khan, Fractional calculus of generalized $p$-$k$-Mittag-Leffler function using Marichev-Saigo-Maeda operators, Arab J. Math. Sci., 25 (2019), 156--168
  • View Article
  • Google Scholar
  • MATH


• [14] M. Kamarujjama, N. U. Khan, O. Khan, The generalized $p$-$k$-Mittag-Leffler function and solution of fractional kinetic equation, J. Anal., 2019 (2019), 18 pages
  • View Article
  • Google Scholar


• [15] M. Kamarujjama, N. U. Khan, O. Khan, J. J. Nieto, Extended type $k$-Mittag-Leffler function and its applications, Int. J. Appl. Comput. Math., 4 (2019), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] O. Khan, N. U. Khan, D. Baleanu, K. S. Nisar, Computable solution of fractional kinetic equations using Mathieu-type series, Adv. Difference Equ., 2019 (2019), 13 pages
  • View Article
  • Google Scholar


• [17] A. A. Kilbas, M. Saigo, Fractional calculus of the H-function, Fukuoka Univ. Sci. Rep., 28 (1998), 41--51
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] A. A. Kilbas, N. Sebastian, Generalized fractional integration of Bessel function of first kind, Integral Transforms Spec. Funct., 19 (2008), 869--883
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] O. L. Marichev, Voletra equation of Mellin convolution type with a horn function in the kernel, Izvestiya Akademii Nauk BSSR Seriya Fiziko-Matematicheskikh Nauk, 1 (1974), 128--129
  • Google Scholar


• [20] E. L. Mathieu, Traité de physique mathématique, Gauthier-Villars, Paris (1980)
  • View Article
  • Google Scholar


• [21] V. N. Mishra, D. L. Suthar, S. D. Purohit, Marichev-Saigo-Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function, Cogent Math., 4 (2017), 11 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] G. Mittag-Leffler, Sur la Nouvelle Fonction $E_{\alpha}(x)$, Comptes Rendus de l'Academie des Sciences Paris, 137 (1903), 554--558
  • Google Scholar
  • MATH


• [23] K. S. Nisar, D. Baleanu, M. M. Al Qurashi, Fractional calculus and application of generalized Struve function, Springer Plus, 2016 (2016), 13 pages
  • View Article
  • Google Scholar


• [24] K. S. Nisar, A. F. Eata, M. Al-Dhaifallah, J. S. Choi, Fractional calculus of generalized k-Mittag-Leffler function and its applications to statistical distribution, Adv. Difference Equ., 2016 (2016), 17 pages
  • View Article
  • Google Scholar


• [25] K. S. Nisar, D. L. Suthar, M. Bohra, S. D. Purohit, Generalized fractional integral operators pertaining to the by-product of Srivastava's polynomials and generalized Mathieu Series, Mathematics, 7 (2019), 8 pages
  • View Article
  • Google Scholar


• [26] T. R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the Kernel, Yokohama Math. J., 19 (1971), 7--15
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] H. Y. Qin, Z. Y. Gu, Y. L. Fu, T. X. Li, Existence of Mild Solutions and Controllability of Fractional Impulsive Integrodifferential Systems with Nonlocal Conditions, J. Funct. Spaces, 2017 (2017), 11 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] G. Rahman, S. Mubeen, K. S. Nisar, J. S. Choi, Certain extended special functions and fractional integral and derivative operators via an extended beta function, Nonlinear Functional Analysis and Applications, 24 (2019), 13 pages
  • View Article
  • Google Scholar


• [29] G. Rahman, K. S. Nisar, J. S. Choi, S. Mubeen, M. Arshad, Pathway Fractional Integral Formulas Involving Extended Mittag-Leffler Functions in the Kernel, Kyungpook Math. J., 59 (2019), 125--134
  • View Article
  • MathSciNet
  • Google Scholar


• [30] M. Saigo, A remark on integral operators involving the gauss hypergeometric functions, Math. Rep. Kyushu Univ., 11 (1977/78), 135--143
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] M. Saigo, N. Maeda, More generalization of fractional calculus, Transform methods and special functions, in: Transform Methods and Special Functions, 1996 (1996), 386--400
  • Google Scholar


• [32] R. K. Saxena, M. Saigo, Certain properties of fractional calculus operators associated with generalized Mittag-Leffler function, Fract. Calc. Appl. Anal., 8 (2005), 141--154
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [33] A. K. Shukla, J. C. Prajapati, On a generalization of Mittag-Leffler function and its Properties, J. Math. Anal. Appl., 336 (2007), 797--811
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] G. Singh, P. Agrwal, S. Araci, M. Acikgoz, Certain fractional calculus formulas involving extended generalized Mathieu series, Adv. Difference Equ., 2018 (2018), 30 pages
  • View Article
  • Google Scholar


• [35] H. M. Srivastava, A contour integral involving Fox's $H$-function, Indian J. Math., 14 (1972), 1--6
  • MathSciNet
  • Google Scholar
  • MATH


• [36] H. M. Srivastava, P. W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood Ltd., Chichester (1985)
  • MathSciNet
  • Google Scholar
  • MATH


• [37] H. M. Srivastava, R. K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput., 118 (2001), 1--52
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [38] H. M. Srivastava, Ž. Tomovski, Some problems and solutions involving Mathieu’ series and its generalizations, JIPAM. J. Inequal. Pure Appl. Math., 5 (2004), 13 pages
  • MathSciNet
  • Google Scholar


• [39] Ž. Tomovski, K. Mehrez, Some families of generalized Mathieu-type power series, Math. Inequal. Appl., 20 (2017), 973--986
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [40] Ž. Tomovski, T. K. Pogány, Integral expressions for Mathieu-type power series and for the Butzer-Flocke-Hauss function, Fract. Calc. Appl. Anal., 14 (2011), 623--634
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [41] Ž. Tomovski, K. Trenčevski, On an open problem of Bai-Ni Guo and Feng Qi, JIPAM. J. Inequal. Pure Appl. Math., 4 (2003), 7 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [42] P. G. Wang, C. R. Li, J. Zhang, T. X. Li, Quasilinearization method for first-order impulsive integro-differential equations, Electron. J. Differential Equations, 2019 (2019), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [43] A. Wiman, Über den fundamental satz in der theorie der funktionen $E_{\alpha}(x)$, Acta Mathematica, 29 (1995), 191--201
  • Google Scholar


• [44] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, J. London Math. Soc., 10 (1935), 257--270
  • View Article
  • Google Scholar