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An extension of the multiple Erdélyi-Kober operator and representations of the generalized hypergeometric functions

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Published/Copyright: January 13, 2019
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 21 Issue 5

Abstract

In this paper we investigate the extension of the multiple Erdélyi-Kober fractional integral operator of Kiryakova to arbitrary complex values of parameters by the way of regularization. The regularization involves derivatives of the function in question and the integration with respect to a kernel expressed in terms of special case of Meijer’s G-function. An action of the regularized multiple Erdélyi-Kober operator on some simple kernels leads to decomposition formulas for the generalized hypergeometric functions. In the ultimate section, we define an alternative regularization better suited for representing the Bessel type generalized hypergeometric function p−1Fp. A particular case of this regularization is then used to identify some new facts about the positivity and reality of zeros of this function.

MSC 2010: Primary 26A33; Secondary 33C60; 33C20
Key Words and Phrases: multiple Erdélyi-Kober operator; Meijer’s G-function; generalized hypergeometric function; Hadamard finite part

Acknowledgements

The research of the first author has been supported by the Russian Science Foundation under the project 14-11-00022. The research of the second author has been supported by the Spanish Ministry of “Economía y Competitividad” under the project MTM2017-83490-P and by the Universidad Pública de Navarra.

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Received: 2018-01-29
Published Online: 2019-01-13
Published in Print: 2018-10-25

© 2018 Diogenes Co., Sofia

Articles in the same Issue

  1. Editorial Note
  2. FCAA related news, events and books (FCAA–volume 21–5–2018)
  3. Discussion Paper
  4. Fractional calculus’s adventures in Wonderland (Round table held at ICFDA 2018)
  5. Survey Paper
  6. On modifications of the exponential integral with the Mittag-Leffler function
  7. Research Paper
  8. Global solutions to stochastic Volterra equations driven by Lévy noise
  9. Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker–Planck–Kolmogorov equations
  10. Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems
  11. Regularity of solutions to space–time fractional wave equations: A PDE approach
  12. Asymptotically periodic solutions for Caputo type fractional evolution equations
  13. Complex fractional Zener model of wave propagation in ℝ
  14. Maximum principles for time-fractional Cauchy problems with spatially non-local components
  15. An extension of the multiple Erdélyi-Kober operator and representations of the generalized hypergeometric functions
  16. A further extension of Mittag-Leffler function
  17. Frequency-distributed representation of irrational linear systems
  18. Centre-of-mass like superposition of Ornstein–Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion
https://doi.org/10.1515/fca-2018-0071
Keywords for this article
multiple Erdélyi-Kober operator; Meijer’s G-function; generalized hypergeometric function; Hadamard finite part

Articles in the same Issue

  1. Editorial Note
  2. FCAA related news, events and books (FCAA–volume 21–5–2018)
  3. Discussion Paper
  4. Fractional calculus’s adventures in Wonderland (Round table held at ICFDA 2018)
  5. Survey Paper
  6. On modifications of the exponential integral with the Mittag-Leffler function
  7. Research Paper
  8. Global solutions to stochastic Volterra equations driven by Lévy noise
  9. Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker–Planck–Kolmogorov equations
  10. Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems
  11. Regularity of solutions to space–time fractional wave equations: A PDE approach
  12. Asymptotically periodic solutions for Caputo type fractional evolution equations
  13. Complex fractional Zener model of wave propagation in ℝ
  14. Maximum principles for time-fractional Cauchy problems with spatially non-local components
  15. An extension of the multiple Erdélyi-Kober operator and representations of the generalized hypergeometric functions
  16. A further extension of Mittag-Leffler function
  17. Frequency-distributed representation of irrational linear systems
  18. Centre-of-mass like superposition of Ornstein–Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion
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