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Generalized Mittag-Leffler Function and Its Properties

Year 2015, Volume: 3 Issue: 3, 12 - 18, 26.06.2015

Abstract

Recently, Srivastava, C¸ etinkaya and Kıymaz [18] defined the generalized Pochhammer symbol and obtained some relations.In this paper, we define the generalized Mittag-Leffler function via the generalized Pochammer symbol and present some recurrencerelation, derivative properties, integral representation. Moreover, we obtain a relation between wright hypergeometric function and thegeneralized Mittag-Leffler function

References

  • Agarwal, R. P: A propos d’ une note de M. Pierre Humbert, Comptes Rendus de l’ Academie des Sciences, vol. 236, pp. 203-2032, 1953.
  • Chaudhry M.A, Srivastava H.M, Paris R.B : Extended hypergeometric and confluent hypergeometric functions, Applied Mathematics and Computation, 159 (2004) 589-602.
  • Chaudhry M.A, Zubair, S.M: On a Class of Incomplete Gamma Functions with Applications. Haubold H. J., Mathai A. M., and Saxena R. K: Mittag-Leffler Functions and Their Applications, Journal of Applied Mathematics, Vol 2011, 51 pages.
  • Humbert P. and Agarwal, R. P : Sur la fonction de Mittag-Leffler et quelques unes de ses generalizations, Bulletin of Science and Mathematics Series II, vol. 77, pp.180-185, 1953.
  • Kurulay M, Bayram M: Some properties of the Mittag-Leffler functions and their relation with the Wright function, Advance Difference Equations 2012, 2012:178.
  • Mittag-Leffler, G. M: Une generalisation de l’ integrale de Laplace-Abel, Comptes Rendus de l’ Academie des Sciences Serie II, vol. 137, pp. 537-539, 1903.
  • Mittag-Leffler,G. M: Sur la nouvelle fonction Eα(x),Comptes Rendus de l’ Academie des Sciences, vol. 137, pp. 554-558, 1903.
  • Mittag-Leffler, G. M, Mittag-Leffler, Sur la representation analytiqie d’une fonction monogene (cinquieme note), Acta Mathematica, vol. 29, no. 1, pp. 101-181, 1905.
  • ¨Ozarslan M.A : Some Remarks on Extended Hypergeometric, Extended Cofluent Hypergeometric and Extended Appell’s Functions, Journal of Compuattional Analysis and Applications, Vol. 14, NO:6, 1148-1153, 2012.
  • ¨Ozarslan M.A, ¨Ozergin E: Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Mathematical and ComputerModelling, 52 (2010) 1825-1833.
  • Prabhakar T. R. : A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), pp. 7-15.
  • Samko, S. G, Kilbas,A. A. and Marichev, O. I: Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, NY, USA, 1993.
  • Srivastava H.M, Manocha, H. L: A Treatise on Generating Functions. Wiman, A :Uber den fundamentalsatz in der theorie der funktionen Eα(x), Acta Math., Vol. 29, p.p. 191-201, 1905.
  • Wiman, A: ¨Uber die Nullstellun der Funktionen Eα(x),Acta Mathematica, vol. 29, pp. 217-234, 1905.
  • ¨Ozarslan, M. A, Yılmaz Yas¸ar, B: The Extended Mittag-Leffler’s Function and Its Properties, Journal of Inequalities and Applications, 2013.
  • Srivastava, H.M, C¸ etinkaya, A, Kıymaz, O: A certain generalized Pochammer symbol and ıts applications to hypergeometric functions, Applied Mathematics and Computation, 226 (2014) 484-491.
Year 2015, Volume: 3 Issue: 3, 12 - 18, 26.06.2015

Abstract

References

  • Agarwal, R. P: A propos d’ une note de M. Pierre Humbert, Comptes Rendus de l’ Academie des Sciences, vol. 236, pp. 203-2032, 1953.
  • Chaudhry M.A, Srivastava H.M, Paris R.B : Extended hypergeometric and confluent hypergeometric functions, Applied Mathematics and Computation, 159 (2004) 589-602.
  • Chaudhry M.A, Zubair, S.M: On a Class of Incomplete Gamma Functions with Applications. Haubold H. J., Mathai A. M., and Saxena R. K: Mittag-Leffler Functions and Their Applications, Journal of Applied Mathematics, Vol 2011, 51 pages.
  • Humbert P. and Agarwal, R. P : Sur la fonction de Mittag-Leffler et quelques unes de ses generalizations, Bulletin of Science and Mathematics Series II, vol. 77, pp.180-185, 1953.
  • Kurulay M, Bayram M: Some properties of the Mittag-Leffler functions and their relation with the Wright function, Advance Difference Equations 2012, 2012:178.
  • Mittag-Leffler, G. M: Une generalisation de l’ integrale de Laplace-Abel, Comptes Rendus de l’ Academie des Sciences Serie II, vol. 137, pp. 537-539, 1903.
  • Mittag-Leffler,G. M: Sur la nouvelle fonction Eα(x),Comptes Rendus de l’ Academie des Sciences, vol. 137, pp. 554-558, 1903.
  • Mittag-Leffler, G. M, Mittag-Leffler, Sur la representation analytiqie d’une fonction monogene (cinquieme note), Acta Mathematica, vol. 29, no. 1, pp. 101-181, 1905.
  • ¨Ozarslan M.A : Some Remarks on Extended Hypergeometric, Extended Cofluent Hypergeometric and Extended Appell’s Functions, Journal of Compuattional Analysis and Applications, Vol. 14, NO:6, 1148-1153, 2012.
  • ¨Ozarslan M.A, ¨Ozergin E: Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Mathematical and ComputerModelling, 52 (2010) 1825-1833.
  • Prabhakar T. R. : A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), pp. 7-15.
  • Samko, S. G, Kilbas,A. A. and Marichev, O. I: Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, NY, USA, 1993.
  • Srivastava H.M, Manocha, H. L: A Treatise on Generating Functions. Wiman, A :Uber den fundamentalsatz in der theorie der funktionen Eα(x), Acta Math., Vol. 29, p.p. 191-201, 1905.
  • Wiman, A: ¨Uber die Nullstellun der Funktionen Eα(x),Acta Mathematica, vol. 29, pp. 217-234, 1905.
  • ¨Ozarslan, M. A, Yılmaz Yas¸ar, B: The Extended Mittag-Leffler’s Function and Its Properties, Journal of Inequalities and Applications, 2013.
  • Srivastava, H.M, C¸ etinkaya, A, Kıymaz, O: A certain generalized Pochammer symbol and ıts applications to hypergeometric functions, Applied Mathematics and Computation, 226 (2014) 484-491.
There are 16 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Banu Yılmaz Yaşar This is me

Publication Date June 26, 2015
Published in Issue Year 2015 Volume: 3 Issue: 3

Cite

APA Yaşar, B. . Y. (2015). Generalized Mittag-Leffler Function and Its Properties. New Trends in Mathematical Sciences, 3(3), 12-18.
AMA Yaşar BY. Generalized Mittag-Leffler Function and Its Properties. New Trends in Mathematical Sciences. June 2015;3(3):12-18.
Chicago Yaşar, Banu Yılmaz. “Generalized Mittag-Leffler Function and Its Properties”. New Trends in Mathematical Sciences 3, no. 3 (June 2015): 12-18.
EndNote Yaşar BY (June 1, 2015) Generalized Mittag-Leffler Function and Its Properties. New Trends in Mathematical Sciences 3 3 12–18.
IEEE B. . Y. Yaşar, “Generalized Mittag-Leffler Function and Its Properties”, New Trends in Mathematical Sciences, vol. 3, no. 3, pp. 12–18, 2015.
ISNAD Yaşar, Banu Yılmaz. “Generalized Mittag-Leffler Function and Its Properties”. New Trends in Mathematical Sciences 3/3 (June 2015), 12-18.
JAMA Yaşar BY. Generalized Mittag-Leffler Function and Its Properties. New Trends in Mathematical Sciences. 2015;3:12–18.
MLA Yaşar, Banu Yılmaz. “Generalized Mittag-Leffler Function and Its Properties”. New Trends in Mathematical Sciences, vol. 3, no. 3, 2015, pp. 12-18.
Vancouver Yaşar BY. Generalized Mittag-Leffler Function and Its Properties. New Trends in Mathematical Sciences. 2015;3(3):12-8.