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Expression of the Holtsmark function in terms of hypergeometric \(_{2}F_2\) and Airy Bi functions

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The Holtsmark distribution has applications in plasma physics, for the electric-microfield distribution involved in spectral line shapes for instance, as well as in astrophysics for the distribution of gravitating bodies. It is one of the few examples of a stable distribution for which a closed-form expression of the probability density function is known. However, the latter is not expressible in terms of elementary functions. In the present work, we mention that the Holtsmark probability density function can be expressed in terms of hypergeometric function \(_{2}F_2\) and of Airy function of the second kind Bi and its derivative. The new formula is simpler than the one proposed by Lee involving \(_{2}F_3\) and \(_{3}F_4\) hypergeometric functions.

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  1. 1.

    J. Holtsmark, Uber die Verbreiterung von Spektrallinien. Ann. Phys. (Leipzig) 58, 577–630 (1919)

  2. 2.

    J.D. Hey, A generalization of some functions of Holtsmark, and Chandrasekhar and Von Neumann. J. Quant. Spectrosc. Radiat. Transf. 16, 931–947 (1976)

  3. 3.

    J.D. Hey, Further properties of the generalized functions of Holtsmark, and Chandrasekhar and Von Neumann. J. Quant. Spectrosc. Radiat. Transf. 41, 167–171 (1989)

  4. 4.

    D. Salzmann, Atomic Physics in Hot Plasmas, International Series of Monographs on Physics (Oxford University Press, Oxford, 1998)

  5. 5.

    A.V. Demura, Physical models of plasma microfield. Int. J. Spectrosc. 671073, 42 (2010)

  6. 6.

    M.A. Gigosos, Stark broadening models for plasma diagnostics. J. Phys. D Appl. Phys. 47, 343001 (2014)

  7. 7.

    S. Chandrasekhar, The statistics of the gravitational field arising from a random distribution of stars. I. The speed of fluctuations. Astrophys. J. 95, 489–531 (1942)

  8. 8.

    S. Chandrasekhar, Stochastic Problems in Physics and Astronomy. Rev. Mod. Phys. 15, 1–89 (1943)

  9. 9.

    L. Pietronero, M. Bottaccio, R. Mohayaee, M. Montuori, The Holtsmark distribution of forces and its role in gravitational clustering. J. Phys. Condens. Matter 14, 2141–2152 (2002)

  10. 10.

    L.G. D’yachkov, Approximation for the probabilities of the realization of atomic bound states in a plasma. J. Quant. Spectrosc. Radiat. Transf. 59, 65–69 (1998)

  11. 11.

    O.E. Barndorff-Nielsen, T. Mikosch, S.I. Resnick, Lévy Processes: Theory and Applications (Springer, Berlin, 2001)

  12. 12.

    E. Stambulchik, Stark effect of high-\(n\) hydrogen-like transitions: quasi-contiguous approximation. J. Phys. B At. Mol. Opt. Phys. 41, 095703 (2008)

  13. 13.

    E. Stambulchik, Y. Maron, Quasi-contiguous approximation for line-shape modeling in plasmas. AIP Conf. Proc. 1438, 203–209 (2012)

  14. 14.

    E. Stambulchik, Y. Maron, Quasicontiguous frequency-fluctuation model for calculation of hydrogen and hydrogenlike Stark-broadened line shapes in plasmas. Phys. Rev. E 87, 053108 (2013)

  15. 15.

    I.A. Kozlitin, Simulating the Holtsmark distribution by the Monte Carlo method. Math. Models Comput. Simul. 3, 58–64 (2011)

  16. 16.

    A. Poquérusse, Simplified rational approximations for Holtsmark and related functions. Eur. Phys. J. D 10, 307–308 (2000)

  17. 17.

    D. Mihalas, Stellar Atmospheres, 2nd edn. (Freeman, San Francisco, 1978)

  18. 18.

    D.G. Hummer, Rational approximations for the Holtsmark distribution, its cumulative and derivative. J. Quant. Spectrosc. Radiat. Transf. 36, 1–5 (1986)

  19. 19.

    R.B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, London, 1973)

  20. 20.

    G.H. Hardy, J.E. Littlewood, Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes, Acta Mathematica, 41, pp.119-196 (1916). (See notes therein for further references to Cahen’s and Mellin’s work, including Cahen’s thesis.)

  21. 21.

    A.P. Prudnikov, YuA Brychkov, O.I. Marichev, Evaluation of integrals and the Mellin transform, Itogi Nauki i Tekhn. Ser. Mat. Anal. 27, 3–146 (1989)

  22. 22.

    A.P. Prudnikov, YuA Brychkov, O.I. Marichev, Evaluation of integrals and the Mellin transform, Itogi Nauki i Tekhn. J. Soviet Math. 54, 1239–1341 (1991)

  23. 23.

    P. Flajolet, X. Gourdon, P. Dumas, Mellin transforms and asymptotics: Harmonic sums. Theor. Comput. Sci. 144, 3–58 (1995)

  24. 24.

    R.B. Paris, D. Kaminsky, Asymptotics and Mellin-Barnes Integrals (Cambridge University Press, Cambridge, 2001)

  25. 25.

    E.W. Barnes, The Maclaurin sum formula. Proc. Lond. Math. Sot. 2, 253–272 (1905)

  26. 26.

    W. H. Lee, Continuous and Discrete Properties of Stochastic Processes, Ph.D. thesis (University of Nottingham, 2010), pp. 37-39

  27. 27.

    E. Çopuroǧlu, T. Mehmetoǧlu, Analytical evaluation of Holtsmark distribution of energies and its role in plasma microfields. J. Sci. Arts 2, 523–528 (2019)

  28. 28.

    Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL (2018)

  29. 29.

    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover publications Inc, New York, 1972). ninth printing, tenth GPO printing

  30. 30.

    A.K. Ghatak, R.L. Gallawa, I.C. Goyal, Modified Airy Function and WKB Solutions to the Wave Equation, NIST Monograph 176 (U.S. Government Printing Office, Washington, 1991)

  31. 31.

    NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/9.6, Release 1.0.24 of 2019-09-15. F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders, H.S. Cohl, M.A. McClain, eds., Eqs. 9.6.25 and 9.6.26

  32. 32.

    A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and series, Vol. 3: More Special Functions (Gordon and Breach, Amsterdam-Paris-New-York, 1986), translated from the Russian: Integraly i ryady by N. M. Queen

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Correspondence to Jean-Christophe Pain.

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Pain, J. Expression of the Holtsmark function in terms of hypergeometric \(_{2}F_2\) and Airy Bi functions. Eur. Phys. J. Plus 135, 236 (2020). https://doi.org/10.1140/epjp/s13360-020-00248-4

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