Erdélyi-Kober Fractional Integrals in the Real Scalar Variable Case

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 31)


This monograph will examine a new definition for fractional integrals in terms of the distributions of products and ratios of statistically independently distributed positive scalar random variables or in terms of Mellin convolutions of products and ratios in the case of real scalar variables. The idea will be generalized to cover real multivariate cases as well as to real matrix-variate cases. In the matrix-variate case, M-convolutions of products and ratios will be used to extend the ideas. Then we will give a definition for the case of real-valued scalar functions of several real matrices. Then we examine fractional calculus in the complex domain. Here p × p Hermitian positive definite matrices and real-valued scalar functions of these matrices are examined to define and evaluate fractional integrals. It is shown that one can define all types of fractional integrals and fractional derivatives through Erdélyi-Kober fractional integral operators and statistical distribution theory. Then differential operators will be defined by using the following argument. If Dα denotes fractional integral of order α then Dα will be called fractional derivative of order α. For n = 1, 2, …, Dn denotes the integer order derivatives. For \(\Re (n-\alpha )>0\) we can define Dα = DnD−(nα) or Dα = D−(nα)Dn. The first will be fractional derivative of order α in the Riemann-Liouville sense and the latter in the Caputo sense. In the present chapter we concentrate on fractional integrals in the real scalar cases.


  1. 1.
    G. Calcagni, Geometry of fractional spaces. Adv. Theor. Math. Phys. 16, 549–644 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Gorenflo, F. Mainardi, Fractional calculus integral and differential equations of fractional order, in Fractal and Fractional Calculus in Continuum Mechanics, ed. by A. Carpinteri, F. Mainardi (Springer, Wien and New York, 1997), pp. 223–276CrossRefGoogle Scholar
  3. 3.
    R. Herrmann, Towards a geometric interpretation of generalized fractional integrals – Erdélyi-Kober type integrals on R(N) as an example. Fract. Calc. Appl. Anal. 17(2), 361–370 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    R.Hilfer (ed.), Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000)zbMATHGoogle Scholar
  5. 5.
    A.A. Kilbas, J.J. Trujillo, Computation of fractional integral via functions of hypergeometric and Bessel type. J. Comput. Appl. Math. 118(1–2), 223–239 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res Notes Math 301, Longman Scientific & Technical: Harlow, Co-published with (Wiley, New York, 1994)Google Scholar
  7. 7.
    V. Kiryakova, Y. Luchko, Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators. Centr. Eur. J. Phys. 11(10), 1314–1336 (2013)Google Scholar
  8. 8.
    D. Kumar, P-transform. Integral Transforms Spec. Funct. 22(8), 603–616 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Kumar, H.J. Haubold, On extended thermonuclear functions through pathway model. Adv. Space Res. 45, 698–708 (2010)CrossRefGoogle Scholar
  10. 10.
    D. Kumar, A.A. Kilbas, Fractional calculus of P-transform. Fract. Calc. Appl. Anal. 13(3), 309–328 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Y. Luchko, Operational rules for a mixed operator of the Erdélyi-Kober type. Fract. Calc. Appl. Analysis 7(3), 339–364 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Y. Luchko, J.J. Trujillo, Caputo-type modification of the Erdélyi-Kober fractional derivative. Fract. Calc. Appl. Anal. 10(3), 250–267 (2007)zbMATHGoogle Scholar
  13. 13.
    F. Mainardi, Yu. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 153–192 (2001)MathSciNetzbMATHGoogle Scholar
  14. 14.
    A.M. Mathai, Pathway to matrix-variate gamma and normal densities. Linear Algebra Appl. 396, 317–328 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    A.M. Mathai, Generalized Krätzel integral and associated statistical densities. Int. J. Math. Anal. 6(51), 2501–2510 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    A.M. Mathai, Fractional integral operators in the complex matrix-variate case. Linear Algebra Appl. 439, 2901–2913 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    A.M. Mathai, Fractional integral operators involving many matrix variables. Linear Algebra Appl. 446, 196–215 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    A.M. Mathai, H.J. Haubold, Special Functions for Applied Scientists (Springer, New York, 2008)CrossRefGoogle Scholar
  19. 19.
    A.M. Mathai, H.J. Haubold, Stochastic processes via pathway model. Entropy 17, 2642–2654 (2015)CrossRefGoogle Scholar
  20. 20.
    A.M. Mathai, S.B. Provost, T. Hayakawa, Bilinear Forms and Zonal Polynomials. Lecture Notes in Statistics (Springer, New York, 1995)Google Scholar
  21. 21.
    A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-function: Theory and Applications (Springer, New York, 2010)CrossRefGoogle Scholar
  22. 22.
    R. Metzler, W.G. Glöckle, T.F. Nonnenmacher, Fractional model equation for anomalous diffusion. Physica A 211, 13–24 (1994)CrossRefGoogle Scholar
  23. 23.
    K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)zbMATHGoogle Scholar
  24. 24.
    K. Nishimoto (1984/1987/1989/1991/1996) Fractional Calculus, vols. 1–5 (Descartes Press, Koriyama, 1996)Google Scholar
  25. 25.
    K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Academic, New York, 1974)zbMATHGoogle Scholar
  26. 26.
    G. Pagnini, Erdélyi-Kober fractional diffusion. Fract. Calc. Appl. Anal. 15(1), 117–127 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    I. Podlubny, Fractional Differential Equations (Academic, San Diego, 1999)zbMATHGoogle Scholar
  28. 28.
    L. Plociniczak, Approximation of the Erdélyi-Kober operator with application to the time-fractional porous medium equation. SIAM J. Appl. Math. 74(4), 129–1237 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    M. Saigo, A.A. Kilbas, Generalized fractional calculus of the H-function. Fukuoka Univ. Sci. Rep. 29, 31–45 (1999)MathSciNetzbMATHGoogle Scholar
  30. 30.
    I.N. Sneddon, The use in mathematical physics of Erdélyi-Kober operators and of some of their generalizations, in Fractional Calculus and Its Applications, ed. by B. Ross (Springer, New York, 1975)Google Scholar
  31. 31.
    H.M. Srivastava, R.K. Saxena, Operators of fractional integration and their applications. Appl. Math. Comput. 118, 1–52 (2001)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Office for Outer Space Affairs United NationsViennaAustria

Personalised recommendations