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EULER TYPE INTEGRAL INVOLVING GENERALIZED MITTAG-LEFFLER FUNCTION

  • Ahmed, Shakeel (Department of Applied Mathematics Faculty of Engineering Aligarh Muslim University) ;
  • Khan, Mumtaz Ahmad (Department of Applied Mathematics Faculty of Engineering Aligarh Muslim University)
  • Received : 2012.05.04
  • Published : 2014.07.31

Abstract

In this paper, we obtain some theorems on certain Euler type integrals involving generalized Mittag-Leffler function. Further, we deduce some special cases involving Wiman function, Prabhakar function and exponential and binomial functions.

Keywords

Euler type integrals;extended beta function;Mittag-Leffler function;Wiman function;Prabhakar function

References

  1. L. C. Andrews, Special Functions of Mathematics for Engineers, SPIE Optical Engineering Press, Bellingham, WA; Oxford University Press, Oxford, 1998.
  2. L. Carlitz, Some extension and convolution formulas related to Mac Mohan's master theorems, SIAM J. Math. Anal. 8 (1977), no. 2, 320-336. https://doi.org/10.1137/0508023
  3. M. A. Chaudhry, A. Qadir, M. Rafiq, and S. M. Zubair, Extension of Euler's beta function, J. Comput. Appl. Math. 78 (1997), no. 1, 19-32. https://doi.org/10.1016/S0377-0427(96)00102-1
  4. M. A. Chaudhry and S. M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math. 55 (1994), no. 1, 99-124. https://doi.org/10.1016/0377-0427(94)90187-2
  5. S. Khan, B. Agrawal, M. A. Pathan, and F. Mohammad, Evaluations of certain Euler type integrals, Appl. Math. Comput. 189 (2007), no. 2, 1993-2003. https://doi.org/10.1016/j.amc.2006.12.073
  6. G. M. Mittag-Leffler, Sur la nouvelle fonction $E_{\alpha}(x)$, C. R. Acad. Sci. Paris 137 (1903), 554-558.
  7. T. R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7-15.
  8. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Matichev, Integrals and Series, vol. 1, Gordan and Breach Science Publishers, New York, 1990.
  9. E. D. Rainville, Special Functions, Macmillan, New York, 1960.
  10. A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl. 336 (2007), no. 2, 797-811. https://doi.org/10.1016/j.jmaa.2007.03.018
  11. A. Wiman, Uber den Fundamental Satz in der Theorie der Funktionen $E_{\alpha}(x)$, Acta Math. 29 (1905), no. 1, 191-201. https://doi.org/10.1007/BF02403202

Cited by

  1. Extension of prefunctions and its relation with Mittag-Leffler function 2017, https://doi.org/10.1007/s41478-017-0052-7