Further results on the generalized Mittag-Leffler function operator

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• 作者：Ram K Saxena (1)
  Jignesh P Chauhan (2)
  Ranjan K Jana (2)
  Ajay K Shukla (2)
  
  1. Department of Mathematics and Statistics ; Jai Narain Vyas University ; Jodhpur ; 342004 ; India
  2. Department of Applied Mathematics & Humanities ; S.V. National Institute of Technology ; Surat ; 395007 ; India
  
• 关键词：33E12 ; 44A10 ; 26A33 ; generalized Mittag ; Leffler function ; Laplace transform ; Mellin transform ; H ; function ; Mellin ; Barnes type integrals ; Riemann ; Liouville fractional integral ; Hilfer derivative
• 刊名：Journal of Inequalities and Applications
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：2015
• 期：1
• 全文大小：904 KB
• 参考文献：1. Mittag-Leffler, GM (1903) Sur la nouvelle fonction E 伪 ( x ) $E_{\alpha} (x )$. C. R. Acad. Sci. Paris 137: pp. 554-558
  2. Mittag-Leffler, GM (1905) Sur la repr茅sentation analytique d鈥檜ne fonction monog猫ne (cinquieme note). Acta Math. 29: pp. 101-181 CrossRef
  3. Wiman, A (1905) 脺ber den Fundamentalsatz in der Theorie der Funktionen E 伪 ( x ) $E_{\alpha} (x )$. Acta Math. 29: pp. 191-201 CrossRef
  4. Prabhakar, TR (1971) A singular integral equation associated with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19: pp. 7-15
  5. Shukla, AK, Prajapati, JC (2007) On generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 336: pp. 797-811 CrossRef
  6. Prajapati, JC, Jana, RK, Saxena, RK, Shukla, AK (2013) Some results on generalized Mittag-Leffler function operator. J. Inequal. Appl. 2013: CrossRef
  7. Wright, EM (1935) The asymptotic expansion of generalized hypergeometric function. J. Lond. Math. Soc. 10: pp. 286-293
  8. Wright, EM (1940) The asymptotic expansion of generalized hypergeometric function. Proc. Lond. Math. Soc. 27: pp. 389-408 CrossRef
  9. Mainardi, F, Pagnini, G (2003) The Wright function as solution of time-fractional diffusion equation. Appl. Math. Comput. 141: pp. 51-62 CrossRef
  10. Kilbas, AA, Saigo, HM, Trujillo, JJ (2002) On the generalized Wright function. Fract. Calc. Appl. Anal. 4: pp. 437-460
  11. Srivastava, HM, Tomovski, Z (2009) Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 211: pp. 198-210 CrossRef
  12. Saxena, RK, Ram, J, Vishnoi, M (2010) Fractional differentiation and fractional integration of the generalized Mittag-Leffler function. J. Indian Acad. Math. 32: pp. 153-162
  13. Saxena, RK, Nishimoto, K (2011) Further results on the generalized Mittag-Leffler functions of fractional calculus. J. Fract. Calc. 40: pp. 29-41
  14. Saxena, RK, Nishimoto, K (2010) N-Fractional calculus of generalized Mittag-Leffler functions. J. Fract. Calc. 37: pp. 43-52
  15. Kiryakova, V (1999) Multiple (multiindex) Mittag-Leffler function related Gelfond and Leontiex operators and Laplace type integrals transforms. Fract. Calc. Appl. Anal. 2: pp. 445-462
  16. Kiryakova, V (2000) Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. Appl. Math. 118: pp. 214-259
  17. Saxena, RK, Kalla, SL, Kiryakova, VS (2003) Relations connecting multiindex Mittag-Leffler functions and Riemann-Liouville fractional calculus. Algebras Groups Geom. 20: pp. 365-385
  18. Hilfer, R (2000) Fractional time evolution. Applications of Fractional Calculus in Physics. World Scientific, Singapore
  19. Kilbas, AA, Srivastava, HM, Trujillo, JJ (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam
  20. Mathai, AM, Saxena, RK, Haubold, HJ (2010) The H-Function: Theory and Applications. Springer, New York CrossRef
  21. Samko, SG, Kilbas, AA, Marichev, OI (1993) Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York
  22. Srivastava, HM, Saxena, RK (2001) Operators of fractional integration and their applications. Appl. Math. Comput. 118: pp. 1-52 CrossRef
  23. Erd茅lyi, A, Magnus, W, Oberhettinger, F, Tricomi, FG (1954) Tables of Integral Transforms. McGraw-Hill, New York
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  26. Gorenflo, R, Luchko, Y, Mainardi, F (1999) Analytic properties and applications of the Wright function. Fract. Calc. Appl. Anal. 2: pp. 383-414
  27. G贸rska, K, Penson, KA, Babusci, D, Dattoli, G, Duchamp, GHE (2012) Operator solutions for fractional Fokker-Planck equations. Phys. Rev. E 85: CrossRef
  28. Babusci, D, Dattoli, G, G贸rska, K: On Mittag-Leffler function and associated polynomials (2012). arXiv:1206.3495
  29. Kilbas, AA, Saigo, M, Saxena, RK (2004) Generalized Mittag-Leffler function and generalized fractional calculus. Integral Transforms Spec. Funct. 15: pp. 31-49 CrossRef
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• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X

文摘

The present paper deals with the study of a generalized Mittag-Leffler function operator. This paper is based on the generalized Mittag-Leffler function introduced and studied by Saxena and Nishimoto (J. Fract. Calc. 37:43-52, 2010). Laplace and Mellin transforms of this new operator are investigated. The results are useful where the Mittag-Leffler function occurs naturally. The boundedness and composition properties of this operator are established. The importance of the derived results further lies in the fact that the results of the generalized Mittag-Leffler function defined by Prabhakar (Yokohama Math. J. 19:7-15, 1971), Shukla and Prajapati (J. Math. Anal. Appl. 336:797-811, 2007), and the multiindex Mittag-Leffler function due to Kiryakova (Fract. Calc. Appl. Anal. 2:445-462, 1999; J. Comput. Appl. Math. 118:214-259, 2000; J. Fract. Calc. 40:29-41, 2011) readily follow as a special case of our findings. Further the results obtained are of general nature and include the results given earlier by Prajapati et al. (J. Inequal. Appl. 2013:33, 2013) and Srivastava and Tomovski (Appl. Math. Comput. 211:198-210, 2009). Some special cases of the established results are also given as corollaries.