Abstract

As recently observed by Bazhlekova and Dimovski [1], the n-th derivative of the 2-parametric Mittag-Leffler function gives a 3-parametric Mittag-Leffler function, known as the Prabhakar function. Following the analogy, the n-th derivatives of the Bessel type functions are obtained, and it turns out that usually they are expressed in terms of the Bessel type functions with the same or more number of indices, up to a matching power function. Further, some special cases of the fractional order Riemann-Liouville derivatives and integrals of the Bessel type functions are calculated and interesting relations are proved.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 32
Issue: 3
Year: 2019

DOI: 10.12732/ijam.v32i3.1

Download Section

Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

1. [1] E. Bazhlekova, I. Dimovski, Exact solution for the fractional cable equation with nonlocal boundary conditions, Central Europ. J. of Phys., 11, No 10 (2013), 1304-1313; DOI: 10.2478/s11534-013-0213-5.
2. [2] E. Bazhlekova, Subordination in a class of generalized time-fractional diffusion-wave equations, Fract. Calc. Appl. Anal., 21, No 4 (2018), 869- 900, DOI: 10.1515/fca-2018-0048.
3. [3] A.A. Kilbas, A.A. Koroleva, Generalized Mittag-Leffler function and its extension, Tr. Inst. Mat. Minsk, 13, No 1 (2005), 43-52 (In Russian).
4. [4] A.A. Kilbas, A.A. Koroleva, Integral transform with the extended generalized Mittag-Leffler function, Math. Modeling and Anal., 11, No 2 (2006), 173-186.
5. [5] A.A. Kilbas, A.A. Koroleva, S.V. Rogosin, Multi-parametric Mittag-Leffler functions and their extension, Fract. Calc. Appl. Anal., 16, No 2 (2013), 378-404; DOI: 10.2478/s13540-013-0024-9.
6. [6] A.A. Kilbas, M. Saigo, R.K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integr. Transf. Spec. Funct., 15, No 1 (2004), 31-49.
7. [7] V. Kiryakova, Generalized Fractional Calculus and Applications, Longman Sci. Tech. & J. Wiley, Harlow - N. York (1994).
8. [8] V. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus, J. of Comput. and Appl. Math., 118 (2000), 241-259, doi: 10.1016/S0377-0427(00)00292-2.
9. [9] V. Kiryakova, The special functions of fractional calculus as generalized fractional calculus operators of some basic functions, Computers and Math. with Appl., 59, No 3 (2010), 1128-1141; doi: 10.1016/j.camwa.2009.05.014.
10. [10] V. Kiryakova, From the hyper-Bessel operators of Dimovski to the generalized FC, Fract. Calc. Appl. Anal., 17, No 4 (2014), 977-1000; DOI: 10.2478/s13540-014-0210-4.
11. [11] V. Kiryakova, Fractional calculus operators of special functions? - The result is well predictable!, Chaos, Solitons & Fractals, 102 (2017), 2-15; doi: 10.1016/j.chaos.2017.03.006.
12. [12] V. Kiryakova, Yu. Luchko, The multi-index Mittag-Leffler functions and their applications for solving fractional order problems in applied analysis, In: American Institute of Physics - Conf. Proc., 1301 (2010), 597-613; doi: 10.1063/1.3526661.
13. [13] Yu. Luchko, Asymptotics of zeros of the Wright function, J. for Analysis and its Applications, 19 (2000), No 1, 1-12.
14. [14] Yu. Luchko, Algorithms for evaluation of the Wright function for the real arguments’ values, Fract. Calc. Appl. Anal., 11, No 1 (2008), 57-75.
15. [15] J.A.T. Machado, V. Kiryakova, The chronicles of Fractional Calculus, Fract. Calc. Appl. Anal., 20, No 2 (2017), 307-336; DOI: 10.1515/fca2017-0017.
16. [16] J. Paneva-Konovska, Multi-index (3m-parametric) Mittag-Leffler functions and fractional calculus, Compt. rend. Acad. bulg. Sci., 64, No 8 (2011), 1089-1098.
17. [17] J. Paneva-Konovska, Periphery behaviour of series in Mittag-Leffler type functions, I, Intern. J. Appl. Math., 29, No 1 (2016), 69-78; doi: 10.12732/ijam.v29i1.6.
18. [18] J. Paneva-Konovska, Periphery behaviour of series in Mittag-Leffler type functions, II, Intern. J. Appl. Math., 29, No 2 (2016), 175-186; doi: 10.12732/ijam.v29i2.2.
19. [19] J. Paneva-Konovska, From Bessel to Multi-Index Mittag Leffler Functions: Enumerable Families, Series in them and Convergence, World Scientific Publ., London (2016); doi: 10.1142/q0026.
20. [20] J. Paneva-Konovska, Differential and integral relations in the class of multiindex Mittag-Leffler functions, Fract. Calc. Appl. Anal., 21, No 1 (2018), 254-265; DOI: 10.1515/fca-2018-0016.
21. [21] J. Paneva-Konovska, Bessel type functions: Relations with integrals and derivatives of arbitrary orders, In: American Institute of Physics - Conf. Proc., 2048 (2018), Art. 050015, 050015-1-050015-6; doi: 10.1063/1.5082114.
22. [22] R.S. Pathak, Certain convergence theorems and asymptotic properties of a generalization of Lommel and Maitland transformations, Proc. Nat. Acad. Sci. India, A-36, No 19 (1966), 81-86.
23. [23] T.R. Prabhakar, A singular equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.
24. [24] A.I. Prieto, S.S. de Romero, H.M. Srivastava, Some fractional-calculus results involving the generalized Lommel-Wright and related functions, Appl. Math. Letters, 20 (2007), 17-22.
25. [25] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, N. York - London (1993).
26. [26] E.M. Wright, On the coeficients of power series having exponential singularities, J. London Math. Soc., 8 (1933), 71-79.
27. [27] E.M. Wright (1940), The generalized Bessel function of order greater than one, Quart. J. Math., Oxford Ser., 11, 36-48.
28. [28] S. Yakubovich, Yu. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions, Kluwer Acad. Publ., Dordrecht - Boston - London (1994).