AIMS Mathematics, 2020, 5(2): 1372-1382. doi: 10.3934/math.2020094

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

S-function associated with fractional derivative and double Dirichlet average

1 Department of Mathematics & Statistics, Jai Narain Vyas University, Jodhpur-342005 (Raj.), India
2 Department of Applied Sciences, College of Agriculture, Sumerpur-Pali, Agriculture University of Jodhpur, Jodhpur-342304 (Raj.), India

The object of this article is to investigate the double Dirichlet averages of S-functions. Representations of such relations are obtained in terms of fractional derivative. Some interesting special cases are also stated.
  Article Metrics


1. O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303.    

2. B. C. Carlson, Lauricella's hypergeometric function FD, J. Math. Anal. Appl., 7 (1963), 452-470.    

3. B. C. Carlson, A connection between elementary and higher transcendental functions, SIAM J. Appl. Math., 17 (1969), 116-148.    

4. B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977.

5. B. C. Carlson, B-splines, hypergeometric functions and Dirichlet average, J. Approx. Theory, 67 (1991), 311-325.    

6. W. Zu Castell, Dirichlet splines as fractional integrals of B-splines, Rocky Mountain J. Math., 32 (2002), 545-559.    

7. J. Daiya, Representing double Dirichlet average in term of K-Mittag-Leffler function associated with fractional derivative, Journal of Chemical, Biological and Physical Sciences, Section C, 6 (2016), 1034-1045.

8. J. Daiya, J. Ram, Dirichlet averages of generalized Hurwitz-Lerch zeta function, Asian J. Math. Comput. Res., 7 (2015), 54-67.

9. R. Diaz, E. Pariguan, On hypergeometric functions and k-Pochhammer symbol, Divulg. Mat., 15 (2007), 179-192.

10. A. Erdélyi, W. Magnus, F. Oberhettinger, et al. Tables of Integral Transforms, Vol. II, McGrawHill, New York-Toronto-London, 1954.

11. S. C. Gupta and B. M. Agrawal, Double Dirichlet average and fractional derivative, Ganita Sandesh, 5 (1991), 47-52.

12. R. K. Gupta, B. S. Shaktawat, D. Kumar, Certain relation of generalized fractional calculus associated with the generalized Mittag-Leffler function, J. Raj. Acad. Phy. Sci., 15 (2016), 117-126.

13. R. K. Gupta, B. S. Shaktawat, D. Kumar, A study of Saigo-Maeda fractional calculus operators associated with the multiparameter K-Mittag-Leffler function, Asian J. Math. Comput. Res., 12 (2016), 243-251.

14. R. K. Gupta, B. S. Shaktawat, D. Kumar, Generalized fractional differintegral operators of the K-series, Honam Math. J., 39 (2017), 61-71.    

15. V. Kiryakova, A brief story about the operators of the generalized fractional calculus, Fract. Calc. Appl. Anal., 11 (2008), 203-220.

16. D. Kumar, On certain fractional calculus operators involving generalized Mittag-Leffler function, Sahand Communication in Mathematical Analysis, 3 (2016), 33-45.

17. D. Kumar, R. K. Gupta, D. S. Rawat, et al. Hypergeometric fractional integrals of multiparameter K-Mittag-Leffler function, Nonlinear Science Letters A: Mathematics, Physics and Mechanics, 9 (2018), 17-26.

18. D. Kumar, S. D. Purohit, Fractional differintegral operators of the generalized Mittag-Leffler type function, Malaya J. Mat., 2 (2014), 419-425.

19. D. Kumar, S. Kumar, Fractional integrals and derivatives of the generalized Mittag-Leffler type function, Internat. Sch. Res. Not., 2014 (2014), 1-5.    

20. Shy-Der Lin, H. M. Srivastava, Some miscellaneous properties and applications of certain operators of fractional calculus, Taiwanese J. Math., 14 (2010), 2469-2495.    

21. P. Massopust, B. Forster, Multivariate complex B-splines and Dirichlet averages, J. Approx. Theory, 162 (2010), 252-269.    

22. E. Neuman, Stolarsky means of several variables, J. Inequal. Pure Appl. Math., 6 (2005), 1-10.

23. E. Neuman, P. J. Van Fleet, Moments of Dirichlet splines and their applications to hypergeometric functions, J. Comput. Appl. Math., 53 (1994), 225-241.    

24. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Translated from the Russian: Integrals and Derivatives of Fractional Order and Some of Their Applications ("Nauka i Tekhnika", Minsk, 1987), Gordon and Breach Science Publishers, UK, 1993.

25. R. K. Saxena, J. Daiya, Integral transforms of the S-function, Le Mathematiche, 70 (2015), 147-159.

26. R. K. Saxena, J. Daiya, A. Singh, Integral transforms of the k-Mittag-Leffler function $E_{k,\alpha ,\beta }^\gamma \left( z \right)$, Le Mathematiche, 69 (2014), 7-16.

27. R. K. Saxena, T. K. Pogány, J. Ram, et al. Dirichlet averages of generalized multi-index MittagLeffler functions, Armen. J. Math., 3 (2010), 174-187.

28. T. Zhang, L. Xiong, Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative, Appl. Math. Lett., 101 (2020), 106072.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved