AUTHORS: Armando Consiglio, Yuri Luchko, Francesco Mainardi
Download as PDF
We start with a short survey of the basic properties of the Wright functions and distinguish between the functions of the first and the second kind. Then we focus on the key role of the Wright functions of the second kind for the probability theory
KEYWORDS: Probability theory, Wright functions, Mittag-Leffler functions, Laplace and Mellin transforms, Completely monotone functions
REFERENCES:
[
1] A. Consiglio & F. Mainardi (2019). On the Evolution of Fractional Diffusive Waves. Ricerche
di Matematica, published on line 06 Dec.
2019. DOI: 10.1007/s11587-019-00476-6
[Eprint: arxiv.org/abs/1910.1259]
WSEAS TRANSACTIONS on MATHEMATICS Armando Consiglio, Yuri Luchko, Francesco Mainardi E-ISSN: 2224-2880 392 Volume 18, 2019
[2] A. Erdelyi, W. Magnus, F. Oberhettinger and F. ´
Tricomi, (1955). Higher Transcendental Functions, 3-rd Volume, McGraw-Hill, New York .
[Bateman Project].
[3] R. Gorenflo, A.A. Kilbas, F. Mainardi & S. Rogosin (2014). Mittag-Leffler Functions. Related
Topics and Applications, Springer, Berlin. Second Edition in preparation.
[4] R. Gorenflo, Yu. Luchko and F. Mainardi (1999).
Analytical Properties and Applications of the
Wright Function. Fract. Calc. Appl. Anal. 2,
383–414.
[5] A. Liemert & A. Klenie (2015). Fundamental
Solution of the Tempered Fractional Diffusion
Equation. J. Math. Phys, 56, 113504.
[6] Yu. Luchko (2000). On the asymptotics of zeros
of the Wright function. Zeitschrift fur Analysis ¨
und ihre Anwendungen, 19, 597–622.
[7] Yu. Luchko (2019). The Wright function and its
applications, in A. Kochubei, Yu.Luchko (Eds.),
Handbook of Fractional Calculus with Applications Vol. 1: Basic Theory, pp. 241- 268.
De Gruyter GmbH, 2019, Berlin/Boston. Series
edited by J. A.Tenreiro Machado.
[8] Yu. Luchko & V. Kiryakova (2013). The Mellin
integral transform in fractional calculus, Fract.
Calc. Appl. Anal. 16, 405–430.
[9] F. Mainardi (1994). On the initial value problem for the fractional diffusion-wave equation,
in: Rionero, S. and Ruggeri, T. (Editors), Waves
and Stability in Continuous Media. World Scientific, Singapore, pp. 246–251.
[Proc. VII-th
WASCOM, Int. Conf. ”Waves and Stability in
Continuous Media”, Bologna, Italy, 4-7 October
1993]
[10] F. Mainardi (1996a). The fundamental solutions
for the fractional diffusion-wave equation. Applied Mathematics Letters, 9 No 6, 23–28.
[11] F. Mainardi (1996b). Fractional relaxationoscillation and fractional diffusion-wave phenomena. Chaos, Solitons & Fractals 7, 1461–
1477.
[12] F. Mainardi (2010). Fractional Calculus and
Waves in Linear Viscoelasticity. Imperial College Press, London and World Scientific, Singapore.
[13] F. Mainardi & A. Consiglio (2020). The Wright
Functions of the Second Kind in Mathematical
Physics. PRE-PRINT submitted to Mathematics.
[14] F. Mainardi, Yu. Luchko and G. Pagnini
(2001). The Fundamental Solution of the SpaceTime Fractional Diffusion Equation. Fract.
Calc. Appl. Anal. 4, 153–192.
[E-print:
arxiv.org/abs/cond-mat/0702419]
[15] F. Mainardi and M. Tomirotti (1997).. Seismic Pulse Propagation with Constant Q
and Stable Probability Distributions. Annali di Geofisica 40, 1311–1328.
[E-print:
arxiv.org/abs/1008.1341]
[16] R.B. Paris (2019). Asymptotics of the special functions of fractional calculus, in: A.
Kochubei, Yu.Luchko (Eds.), Handbook of
Fractional Calculus with Applications, Vol.
1: Basic Theory, pp. 297-325. De Gruyter
GmbH, 2019 Berlin/Boston. Series edited by J.
A.Tenreiro Machado.
[17] A. Saa, & R. Venegeroles (2011). Alternative
numerical computation of one-sided Levy and ´
Mittag-Leffler distributions, Phys. Rev. E 84,
026702.
[18] B. Stankovicˇ (1970). On the function of
E.M. Wright. Publ. de lInstitut Mathematique, ´
Beograd, Nouvelle Ser. ´ 10, 113–124.
[19] R. Wong & Y.-Q. Zhao (1999a). Smoothing of
Stokes’ discontinuity for the generalized Bessel
function. Proc. R. Soc. London A455, 1381–
1400.
[20] R. Wong & Y.-Q. Zhao (1999b). Smoothing of
Stokes’ discontinuity for the generalized Bessel
function II, Proc. R. Soc. London A 455, 3065–
3084.
[21] E.M. Wright (1933). On the coefficients of
power series having exponential singularities.
Journal London Math. Soc., 8, 71–79.
[22] E.M. Wright (1935). The asymptotic expansion
of the generalized Bessel function. Proc. London
Math. Soc. (Ser. II) 38, 257–270.
[23] E.M. Wright, (1940). The generalized Bessel
function of order greater than one. Quart. J.
Math., Oxford Ser. 11, 36–48.