# American Institute of Mathematical Sciences

## Note on a $k$-generalised fractional derivative

 1 The IIS University, Jaipur, Rajasthan-302020, India 2 Manipal University Jaipur, Rajasthan-303007, India

Corresponding author: ekta.jaipur@gmail.com; sunil.joshi@jaipur.manipal.edu

Received  May 2018 Revised  August 2018 Published  March 2019

In this paper, we introduce the $k$-generalised fractional derivatives with three parameters which reduced to $k$-fractional Hilfer derivatives and $k$-Riemann-Liouville fractional derivative as an interesting special cases. Further, we have also introduced some presumably new fascinating results which include the image power function, Laplace transform and composition of $k$-Riemann-Liouville fractional integral with generalized composite fractional derivative. The technique developed in this paper can be used in other situation as well.

Citation: Ekta Mittal, Sunil Joshi. Note on a $k$-generalised fractional derivative. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020045
##### References:
 [1] R. Dıaz and E. Pariguan, On hypergeometric functions and pochhammer k-symbol, Divulg. Mat, 15 (2007), 179-192. Google Scholar [2] G. A. Dorrego and R. A. Cerutti, The k-fractional hilfer derivative, International Journal of Mathematical Analysis, 7 (2013), 543-550. doi: 10.12988/ijma.2013.13051. Google Scholar [3] R. Garra, R. Gorenflo, F. Polito and Ž Tomovski, Hilfer–prabhakar derivatives and some applications, Applied Mathematics and Computation, 242 (2014), 576-589. doi: 10.1016/j.amc.2014.05.129. Google Scholar [4] R. Hilfer et al., Applications of Fractional Calculus in Physics, vol. 35, World Scientific, 2000. doi: 10.1142/9789812817747. Google Scholar [5] O. S. Iyiola, Solving k-fractional hilfer differential equations via combined fractional integral transform methods, British Journal of Mathematics & Computer Science, 4 (2014), 1427-1436. doi: 10.9734/BJMCS/2014/9444. Google Scholar [6] C. G. Kokologiannaki, Properties and inequalities of generalized k-gamma, beta and zeta functions, Int. J. Contemp. Math. Sciences, 5 (2010), 653-660. Google Scholar [7] M. Mansour, Determining the k-generalized gamma function $\gamma$k (x) by functional equations, Int. J. Contemp. Math. Sciences, 4 (2009), 1037-1042. Google Scholar [8] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993. Google Scholar [9] G. Mridula, P. Manohar, L. Chanchalani and A. Subhash, On generalized composite fractional derivative, Walailak Journal of Science and Technology (WJST), 11 (2014), 1069-1076. Google Scholar [10] S. Mubeen and G. Habibullah, k-fractional integrals and application, Int. J. Contemp. Math. Sci, 7 (2012), 89-94. Google Scholar [11] S. Mubeen, A. Rehman and F. Shaheen, Properties of k-gamma, k-beta and k-psi functions, Bothalia Journal, 5 (2014), 371-379. Google Scholar [12] L. G. Romero, L. L. Luque, G. A. Dorrego and R. A. Cerutti, On the k-riemann-liouville fractional derivative, Int. J. Contemp. Math. Sci, 8 (2013), 41-51. doi: 10.12988/ijcms.2013.13004. Google Scholar

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##### References:
 [1] R. Dıaz and E. Pariguan, On hypergeometric functions and pochhammer k-symbol, Divulg. Mat, 15 (2007), 179-192. Google Scholar [2] G. A. Dorrego and R. A. Cerutti, The k-fractional hilfer derivative, International Journal of Mathematical Analysis, 7 (2013), 543-550. doi: 10.12988/ijma.2013.13051. Google Scholar [3] R. Garra, R. Gorenflo, F. Polito and Ž Tomovski, Hilfer–prabhakar derivatives and some applications, Applied Mathematics and Computation, 242 (2014), 576-589. doi: 10.1016/j.amc.2014.05.129. Google Scholar [4] R. Hilfer et al., Applications of Fractional Calculus in Physics, vol. 35, World Scientific, 2000. doi: 10.1142/9789812817747. Google Scholar [5] O. S. Iyiola, Solving k-fractional hilfer differential equations via combined fractional integral transform methods, British Journal of Mathematics & Computer Science, 4 (2014), 1427-1436. doi: 10.9734/BJMCS/2014/9444. Google Scholar [6] C. G. Kokologiannaki, Properties and inequalities of generalized k-gamma, beta and zeta functions, Int. J. Contemp. Math. Sciences, 5 (2010), 653-660. Google Scholar [7] M. Mansour, Determining the k-generalized gamma function $\gamma$k (x) by functional equations, Int. J. Contemp. Math. Sciences, 4 (2009), 1037-1042. Google Scholar [8] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993. Google Scholar [9] G. Mridula, P. Manohar, L. Chanchalani and A. Subhash, On generalized composite fractional derivative, Walailak Journal of Science and Technology (WJST), 11 (2014), 1069-1076. Google Scholar [10] S. Mubeen and G. Habibullah, k-fractional integrals and application, Int. J. Contemp. Math. Sci, 7 (2012), 89-94. Google Scholar [11] S. Mubeen, A. Rehman and F. Shaheen, Properties of k-gamma, k-beta and k-psi functions, Bothalia Journal, 5 (2014), 371-379. Google Scholar [12] L. G. Romero, L. L. Luque, G. A. Dorrego and R. A. Cerutti, On the k-riemann-liouville fractional derivative, Int. J. Contemp. Math. Sci, 8 (2013), 41-51. doi: 10.12988/ijcms.2013.13004. Google Scholar
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