WSEAS Transactions on Mathematics


Print ISSN: 1109-2769
E-ISSN: 2224-2880

Volume 18, 2019

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of WSEAS Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.


Volume 18, 2019



Some Notes on the Wright Functions in Probability Theory

AUTHORS: Armando Consiglio, Yuri Luchko, Francesco Mainardi

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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.

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[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.

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[17] A. Saa, & R. Venegeroles (2011). Alternative numerical computation of one-sided Levy and ´ Mittag-Leffler distributions, Phys. Rev. E 84, 026702.

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WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 18, 2019, Art. #47, pp. 389-393


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