Research article

On generalized $\mathtt{k}$-fractional derivative operator

  • Received: 11 November 2019 Accepted: 03 February 2020 Published: 19 February 2020
  • MSC : 33C05, 33C15

  • The principal aim of this paper is to introduce $\mathtt{k}$-fractional derivative operator by using the definition of $\mathtt{k}$-beta function. This paper establishes some results related to the newly defined fractional operator such as the Mellin transform and the relations to $\mathtt{k}$-hypergeometric and $\mathtt{k}$-Appell's functions. Also, we investigate the $\mathtt{k}$-fractional derivative of $\mathtt{k}$-Mittag-Leffler and the Wright hypergeometric functions.

    Citation: Gauhar Rahman, Shahid Mubeen, Kottakkaran Sooppy Nisar. On generalized $\mathtt{k}$-fractional derivative operator[J]. AIMS Mathematics, 2020, 5(3): 1936-1945. doi: 10.3934/math.2020129

    Related Papers:

  • The principal aim of this paper is to introduce $\mathtt{k}$-fractional derivative operator by using the definition of $\mathtt{k}$-beta function. This paper establishes some results related to the newly defined fractional operator such as the Mellin transform and the relations to $\mathtt{k}$-hypergeometric and $\mathtt{k}$-Appell's functions. Also, we investigate the $\mathtt{k}$-fractional derivative of $\mathtt{k}$-Mittag-Leffler and the Wright hypergeometric functions.


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    [1] P. Agarwal, M. Jleli, M. Tomar, Certain Hermite-Hadamard type inequalities via generalized k-fractional integrals, J. Inequal. Appl., 2017 (2017), 55.
    [2] B. Acay, E. Bas, T. Abdeljawad, Non-local fractional calculus from different view point generated by truncated M-derivative, J. Comput. Appl. Math., 366 (2020), 112410.
    [3] B. Acay, E. Bas, T. Abdeljawad, Fractional economic models based on market equilibrium in the frame of different type kernels, Chaos Soliton. Fract., 130 (2020), 109438.
    [4] E. Bas, R. Ozarslan, D. Baleanu, Comparative simulations for solutions of fractional Sturm-Liouville problems with non-singular operators, Adv. Differ. Equ., 2018 (2018), 350.
    [5] R. Diaz, E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divulgaciones Math., 15 (2007), 179-192.
    [6] G. A. Dorrego, R. A. Cerutti, The k-Mittag-Leffler function, Int. J. Contemp. Math. Sci., 7 (2012), 705-716.
    [7] G. Farid, G. M. Habullah, An extension of Hadamard fractional integral, Int. J. Math. Anal., 9 (2015), 471-482. doi: 10.12988/ijma.2015.5118
    [8] G. Farid, A. U. Rehman, M. Zahra, On Hadamard-type inequalities for k-fractional integrals, Konuralp J. Math., 4 (2016), 79-86.
    [9] S. Habib, S. Mubeen, M. N. Naeem, et al. Generalized k-fractional conformable integrals and related inequalities, AIMS Mathematics, 4 (2019), 343-358. doi: 10.3934/math.2019.3.343
    [10] C. J. Huang, G. Rahman, K. S. Nisar, et al. Some inequalities of the Hermite-Hadamard type for k-fractional conformable integrals, Aust. J. Math. Anal. Appl., 16 (2019), 7.
    [11] S. Mubeen, S. Iqbal, Grüss type integral inequalities for generalized Riemann-Liouville kfractional integrals, J. Inequal. Appl., 2016 (2016), 109.
    [12] S. Iqbal, S. Mubeen, M. Tomar, On Hadamard k-fractional integrals, J. Fract. Calc. Appl., 9 (2018), 255-267.
    [13] C. G. Kokologiannaki, Properties and inequalities of generalized k-gamma, beta and zeta functions, Int. J. Contemp. Math. Sci., 5 (2010), 653-660.
    [14] V. Krasniqi, A limit for the k-gamma and k-beta function, Int. Math. Forum., 5 (2010), 1613-1617.
    [15] A. A. Kilbas, H. M. Sarivastava, J. J. Trujillo, Theory and Application of Fractional Differential Equation, Elsevier Sciences B.V., Amsterdam, 2006.
    [16] M. Mansour, Determining the k-generalized gamma function Γk(x) by functional equations, Int. J. Contemp. Math. Sci., 4 (2009), 1037-1042.
    [17] F. Merovci, Power product inequalities for the Γk function, Int. J. Math. Anal., 4 (2010), 1007-1012.
    [18] S. Mubeen, k-Analogue of Kummer's first formula, J. Inequal. Spec. Funct., 3 (2012), 41-44.
    [19] S. Mubeen, Solution of some integral equations involving confluent k-hypergeometric functions, Appl. Math., 4 (2013), 9-11. doi: 10.4236/am.2013.47A003
    [20] S. Mubeen, G. M. Habibullah, An integral representation of k-hypergeometric functions, Int. Math. Forum, 7 (2012), 203-207.
    [21] S. Mubeen, G. M. Habibullah, k-fractional integrals and application, Int. J. Contemp. Math. Sci., 7 (2012), 89-94.
    [22] S. Mubeen, S. Iqbal, G. Rahman, Contiguous function relations and an integral representation for Appell k-series F1,k, Int. J. Math. Res., 4 (2015), 53-63. doi: 10.18488/journal.24/2015.4.2/24.2.53.63
    [23] S. Mubeen, M. Naz, M, G. Rahman, A note on k-hypergeometric differential equations, J. Inequal. Spec. Funct., 4 (2013), 38-43.
    [24] S. Mubeen, S. Iqbal, Z. Iqbal, On Ostrowski type inequalities for generalized k-fractional integrals, J. Inequ. Spec. Funct., 8 (2017), 3.
    [25] K. S. Nisar, G. Rahman, J. Choi, et al. Certain Gronwall type inequalities associated with riemann-liouville k- and hadamard k-fractional derivatives and their applications, East Asian Math. J., 34 (2018), 249-263.
    [26] F. Qi, G. Rahman, S. M. Hussain, et al. Some inequalities of Chebyšev Type for conformable k-Fractional integral operators, Symmetry, 10 (2018), 614.
    [27] E. D. Rainville, Special Functions, The Macmillan Company, New York, 1960.
    [28] M. Samraiz, E. Set, M. Hasnain, et al. On an extension of Hadamard fractional derivative, J. Inequal. Appl., 2019 (2019), 263.
    [29] E. Set, M. A. Noor, M. U. Awan, et al. Generalized Hermite-Hadamard type inequalities involving fractional integral operators, J. Inequal. Appl., 2017 (2017), 169.
    [30] G. Rahman, K. S. Nisar, A. Ghaffar, et al. Some inequalities of the Grüss type for conformable k-fractional integral operators, RACSAM, 114 (2020), 9.
    [31] M. Tomar, S. Mubeen, J. Choi, Certain inequalities associated with Hadamard k-fractional integral operators, J. Inequal. Appl., 2016 (2016), 234.
    [32] D. Valerio, J. J. Trujillo, M. Rivero, Fractional calculus: A survey of useful formulas, Eur. Phys. J. Spec. Top., 222 (2013), 1827-1846. doi: 10.1140/epjst/e2013-01967-y
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