On Some Special Functions for Conformable Fractional Integrals

Abstract

In this paper, we introduce the $\left( \alpha ,k\right) $-gamma function$,\ \left( \alpha ,k\right) $-beta function, Pochhammer symbol $\left( x\right) _{n,k}^{\alpha }\ $and Laplace transforms for conformable fractional integrals. We prove several properties generalizing those satisfied by the classical gamma function, beta function and Pochhammer symbol. The results presented here would provide generalizations of those given in earlier works.                                                                                                                                                                                                                                                      

Keywords

• Gamma function
• beta function
• Pochhammer symbol
• confromable fractional integrals

References

1. [1] T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57–66.
2. [2] T. Abdeljawad, M. A. Horani and R. Khalil, Conformable fractional semigroup operators, Journal of Semigroup Theory and Applications vol. 2015 (2015) Article ID. 7.
3. [3] G.E. Andrews, R. Askey and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications 71, Cambrige University, 1999.
4. [4] A. Atangana, D. Baleanu, and A. Alsaedi, New properties of conformable derivative, Open Math. 2015; 13: 889–898.
5. [5] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974.
6. [6] R. Diaz and C. Teruel, q;k-Generalized gamma and beta functions, J. Nonlinear Math. Phys., 12 (2005), 118-134.
7. [7] R. Diaz and E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divulgaciones Matematicas Vol. 15 No. 2(2007), pp. 179-192.
8. [8] R. Diaz, C. Ortiz and E. Pariguan, On the k-gamma q-distribution, Cent. Eur. J. Math., 8 (2010), 448-458.

9. [9] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.
10. [10] R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational Apllied Mathematics, 264 (2014), 65-70.
11. [11] S. Mubeen and G. M Habibullah, k-Fractional integrals and application, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 2, 89 - 94.
12. [12] O.S. Iyiola and E.R.Nwaeze, Some new results on the new conformable fractional calculus with application using D’Alambert approach, Progr. Fract. Differ. Appl., 2(2), 115-122, 2016.
13. [13] M. A. Hammad and R. Khalil, Conformable fractional heat differential equations, International Journal of Differential Equations and Applications 13( 3), 2014, 177-183.
14. [14] M. A. Hammad and R. Khalil, Abel’s formula and wronskian for conformable fractional differential equations, International Journal of Differential Equations and Applications 13( 3), 2014, 177-183.
15. [15] U. N. Katugampola,New approach to generalized fractional integral, Appl. Math. Comput. 218 (2011), 860-865.
16. [16] U. N. Katugampola, A new approach to generalized fractional derivatives, Bul. Math. Anal.Appl., 6 (4) (2014), 1-15.
17. [17] U. N. Katugampola, A new fractional derivative with classical properties, e-print arXiv:1410.6535.