A Note on Hadamard Fractional Differential Equations with Varying Coefficients and Their Applications in Probability

In this paper, we show several connections between special functions arising from generalized Conway-Maxwell-Poisson (COM-Poisson) type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions.

[1]  L. Beghin,et al.  Fractional Poisson processes and related planar random motions , 2009 .

[2]  R. Herrmann.  Generalization of the fractional poisson distribution , 2015, 1503.07187.

[3]  Federico Polito,et al.  The space-fractional Poisson process , 2011, 1107.2874.

[4]  M. Meerschaert,et al.  The Fractional Poisson Process and the Inverse Stable Subordinator , 2010, 1007.5051.

[5]  E. Scalas,et al.  A Generalization of the Space-Fractional Poisson Process and its Connection to some Lévy Processes , 2015, 1502.03115.

[6]  V. Kiryakova.  Generalized Fractional Calculus and Applications , 1993 .

[7]  S. Gerhold.  Asymptotics for a variant of the Mittag–Leffler function , 2011, 1103.2285.

[8]  Dumitru Baleanu,et al.  Caputo-type modification of the Hadamard fractional derivatives , 2012, Advances in Difference Equations.

[9]  On a generalized three-parameter wright function of Le Roy type , 2017, 2007.05925.

[10]  J. Hadamard,et al.  Essai sur l'étude des fonctions données par leur développement de Taylor , 1892 .

[11]  Tomoaki Imoto,et al.  A generalized Conway-Maxwell-Poisson distribution which includes the negative binomial distribution , 2014, Appl. Math. Comput..

[12]  A. I. Solomon,et al.  MITTAG-LEFFLER COHERENT STATES , 1999 .

[13]  E. Shishkina,et al.  Fractional powers of Bessel operators , 2017, Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics.

[14]  R. Gorenflo,et al.  A fractional generalization of the Poisson processes , 2007, math/0701454.

[15]  N. Balakrishnan,et al.  A class of weighted Poisson processes , 2008 .

[17]  Anatoly A. Kilbas,et al.  HADAMARD-TYPE FRACTIONAL CALCULUS , 2001 .

[18]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[19]  R. Gorenflo,et al.  Mittag-Leffler Functions, Related Topics and Applications , 2014, Springer Monographs in Mathematics.

[20]  S. Porwal,et al.  On Mittag-Leffler type Poisson distribution , 2017 .

[21]  Subrata Chakraborty,et al.  Mittag - Leffler function distribution - a new generalization of hyper-Poisson distribution , 2014 .

[22]  Ahmed Alsaedi,et al.  Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities , 2017 .

[23]  On Selection of Analog Volcanoes , 2011 .

[24]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[25]  K. Diethelm.  Mittag-Leffler Functions , 2010 .

[26]  Federico Polito,et al.  On some operators involving Hadamard derivatives , 2013, 1304.1058.

[27]  R. Gorenflo,et al.  Analytical properties and applications of the Wright function , 2007, math-ph/0701069.

[28]  Asymptotic results for a multivariate version of the alternative fractional Poisson process , 2016, 1607.04490.

[29]  Ram K. Saxena,et al.  Analytical solution of space-time fractional telegraph-type equations involving Hilfer and Hadamard derivatives , 2016 .

[30]  Francesco Mainardi,et al.  Fractional relaxation with time-varying coefficient , 2014 .

[31]  Enzo Orsingher,et al.  Counting processes with Bernštein intertimes and random jumps , 2015, J. Appl. Probab..

[32]  L. Beghin,et al.  Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator , 2016, Stochastic Analysis and Applications.

[33]  N. Laskin.  Fractional Poisson process , 2003 .