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Comments on the properties of Mittag-Leffler function

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Abstract

The properties of Mittag-Leffler function are reviewed within the framework of an umbral formalism. We take advantage from the formal equivalence with the exponential function to define the relevant semigroup properties. We analyse the relevant role in the solution of Schrödinger type and heat-type fractional partial differential equations and explore the problem of operatorial ordering finding appropriate rules when non-commuting operators are involved. We discuss the coherent states associated with the fractional Schödinger equation, analyze the relevant Poisson type probability amplitude and compare with analogous results already obtained in the literature.

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References

  1. G. Dattoli, S. Licciardi, R.M. Pidatella, arXiv:1702.08520[math.CA] (2017)

  2. F.M. Cholewinski, J.A. Reneke, Electron. J. Differ. Equ. 2003, 1 (2003)

    Google Scholar 

  3. D. Babusci, G. Dattoli, M. Del Franco, Lectures on mathematical methods for physics, RT/2010/58/ENEA, 2010

  4. L.C. Andrews, Special functions for applied mathematicians and engineers (McMillan, 1985)

  5. M.G. Mittag-Leffler, C. R. Acad. Sci. 136, 537 (1903)

    Google Scholar 

  6. R. Gorenflo, A.A. Kilbas, S.V. Rogosin, Integral Transform. Spec. Funct. 7, 215 (1998)

    Article  MathSciNet  Google Scholar 

  7. K.B. Oldham, J. Spanier, in The fractional calculus: theory and applications of differentiation and integration to arbitrary order. Mathematics in science and engineering (Elsevier Science, 1974), Vol. 111

  8. G. Dattoli, E. Di Palma, E. Sabia, K. Górska, A. Horzela, K.A. Penson, Int. J. Appl. Comput. Math. 3, 3489 (2017)

    Article  MathSciNet  Google Scholar 

  9. E.M. Wright, Proc. London Math. Soc. s2-38, 257 (1935)

    Article  Google Scholar 

  10. I. Saichev, G.M. Zaslavsky, Chaos 7, 753 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  11. K. Gorska, K.A. Penson, D. Babusci, G. Dattoli, G.H.E. Duchamp, Phys. Rev. E 85, 031138 (2012)

    Article  ADS  Google Scholar 

  12. G. Dattoli, J.C. Gallardo, A. Torre, Riv. Nuovo Cimento 11, 1 (1988)

    Article  MathSciNet  Google Scholar 

  13. W.H. Louisell, Quantum statistical properties of radiation (John Wiley & Sons Canada, Limited, 1973)

  14. M. Naber, J. Math. Phys. 45, 3339 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  15. N. Laskin, Phys. Rev. E 66, 056108 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  16. N. Laskin, Commun. Nonlinear Sci. Numer. Simul. 8, 201 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  17. N. Laskin, J. Math. Phys. 50, 113513 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  18. V. Uchaikin, R. Sibatov, Fractional kinetics in solids: anomalous charge transport in semiconductors, dieletrics and nanosystems (World Scientific, 2013), par. 1.3.2

  19. D. Babusci, G. Dattoli, arXiv:1112.1570[math-ph] (2011)

  20. W. Magnus, Commun. Pure Appl. Math. 7, 649 (1954)

    Article  Google Scholar 

  21. F.J. Dyson, Phys. Rev. 75, 486 (1949)

    Article  ADS  Google Scholar 

Download references

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Authors and Affiliations

  1. ENEA – Frascati Research Center, Via Enrico Fermi 45, 00044, Frascati, Rome, Italy

    G. Dattoli & S. Licciardi

  2. H.Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, ul.Eliasza – Radzikowskiego 152, 31342, Krakow, Poland

    K. Gorska & A. Horzela

  3. Dep. of Mathematics and Computer Science, University of Catania, Viale A. Doria 6, 95125, Catania, Italy

    S. Licciardi & R. M. Pidatella

Authors
  1. G. Dattoli
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  2. K. Gorska
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  3. A. Horzela
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  4. S. Licciardi
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  5. R. M. Pidatella
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Corresponding author

Correspondence to S. Licciardi.

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Dattoli, G., Gorska, K., Horzela, A. et al. Comments on the properties of Mittag-Leffler function. Eur. Phys. J. Spec. Top. 226, 3427–3443 (2017). https://doi.org/10.1140/epjst/e2018-00073-1

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  • DOI: https://doi.org/10.1140/epjst/e2018-00073-1

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