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March  2020, 13(3): 797-804. doi: 10.3934/dcdss.2020045

Note on a $ k $-generalised fractional derivative

Ekta Mittal 1,, and  Sunil Joshi 2,,

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.

Keywords: $ k $-Gamma function, $ k $-Beta function, $ k $-Gauss hypergeometric function, $ k $-Riemann-Liouville fractional integral operator, $ k $-Riemann-Liouville fractional derivative operator.
Mathematics Subject Classification: 33B15, 33C05, 26A33.
Citation: Ekta Mittal, Sunil Joshi. Note on a $ k $-generalised fractional derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 797-804. 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

show all references

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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