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

Gauhar Rahman; Shahid Mubeen; Kottakkaran Sooppy Nisar; Gauhar Rahman; Shahid Mubeen; Kottakkaran Sooppy Nisar

doi:10.3934/math.2020129

AIMS Mathematics

2020, Volume 5, Issue 3: 1936-1945. doi: 10.3934/math.2020129

Previous Article Next Article

Research article

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

1.
Department of Mathematics, Shaheed Benazir Bhutto University, Sharingal, Pakistan
2.
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
3.
Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, 11991, Prince Sattam bin Abdulaziz University, Kingdom of Saudi Arabia

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.
- beta function,
- $\mathtt{k}$-beta function,
- hypergeometric function,
- $\mathtt{k}$-hypergeometric function,
- Mellin transform,
- fractional derivative,
- Appell's function,
- $\mathtt{k}$-Mittag-Leffler function
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:

Abstract

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.

References

[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 Γ_k(x) by functional equations, Int. J. Contemp. Math. Sci., 4 (2009), 1037-1042.
[17]	F. Merovci, Power product inequalities for the Γ_k 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 F_1,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

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)