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Some applications of the fractional Poisson probability distribution
4.J. Stirling, Methodus Differentialis: Sive Tractatus de Summatione et Interpolatione Serierum Infinitarum (Gul. Bowyer, London, 1730)
4.English translation by I. Tweddle, James Stirling's Methodus Differentialis: An Annotated Translation of Stirling's Text (Springer, London, 2003).
5.C. A. Charalambides, Enumerative Combinatorics (Chapman and Hall, London/CRC, Boca Raton, FL, 2002), Chap. 8.
8.Higher Transcendental Functions, edited by A. Erdélyi (McGraw-Hill, New York, 1955), Vol. 3, Chap. 18, pp. 206–227.
9.J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).
10.R. J. Glauber, Quantum Theory of Optical Coherence, Selected Papers and Lectures (Wiley-VCH, Weinheim, 2007).
14.G. Dobinski, Archiv der Mathematik und Physik, Greifswald 61, 333 (1877).
16.The relationship between Bell numbers and the diagonal matrix element of the power of the number operator on the basis of the standard coherent states at has been found in Ref. 23.
17.L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
18.Stirling numbers, introduced by Stirling (Ref. 4) in 1730, have been studied in the past by many celebrated mathematicians. Among them are Euler, Lagrange, Laplace, and Cauchy. Stirling numbers play an important role in combinatorics, number theory, probability, and statistics. There are two common sets of Stirling numbers, they are the so-called Stirling numbers of the first kind and Stirling numbers of the second kind (for details, see Refs. 5 and 6).
19.Stirling Numbers of the Second Kind, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed., edited by M. Abramowitz, and I. A. Stegun (Dover, New York, 1972), Sec. 24.1.4, pp. 824–825.
20.Bernoulli and Euler Polynomials and the Euler-Maclarin Formula, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed., edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1972), Sec. 23.1, p. 804.
21.The mean [Eq. (41)] and the second order moment were first obtained by Laskin [see Eqs. (26) and (27) in Ref. 1]. The second order moment defined by Eq. (42) can be presented as Eq. (27) of Ref. 1 if we take into account the well-known equations for the gamma function , , and .
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