Special Functions for Applied Scientists

By Arak M. Mathai, Hans J. Haubold. 2008

480 citations recorded by the source corpus.

Subfields: Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6bf1059ec6b0452f467ff77c904696be9fd942cd

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In Statistical and Nonlinear Physics

Citations in the corpus

A fractional counting process and its connection with the Poisson process (2015, unknown): “…∑ n=0 (λ wtν) n E n+1ν,νn+1 (−λ tν) = Eν,1 (λ (w− 1)tν) (|w| ≤ 1, t > 0), we have P { H j ≤ t } = λ jt ν Γ(ν) ∫ 1 0 (1− u)ν−1 Eν,1 (−λ j tνuν)du By making use of formula (2.2.14) of Mathai and Haubold (2008), i.e. ∫ 1 0 zβ−1 (1− z)σ−1 Eα ,β (xzα)dz = Γ(σ)Eα ,σ+β (x) , (where α > 0; β ,σ ∈C; Re(β…”
On some new properties of fractional derivatives with Mittag-Leffler kernel (2017, background): “nel is more easily used from the numerical viewpoint, and this has been studied for example in [15]. We note that the Mittag-Leﬄer function is already known to be highly useful in fractional calculus [26]. It has also been an important part of the deﬁnitions of certain other fractional diﬀerintegral”
Mittag-Leffler Functions and Their Applications (2009, unknown): “9.2”
The characteristic function of rough Heston models (2016, unknown): “The following properties of fα,λ and Fα,λ can be found in [21, 34, 36].”
Science metrics on fractional calculus development since 1966 (2013, background): “Figure 2 (left) depicts the evolution of Da [19, 12, 56, 75, 57, 99, 60, 87, 100, 100, 77, 68, 85, 95, 69, 70, 71, 72, 78, 23, 102, 13, 96, 64, 114, 36, 79, 73, 91, 63, 84, 80, 105, 109, 4, 111, 82, 22, 27, 28, 110, 81, 67, 32, 29, 86, 52, 33, 117, 106, 55, 58, 112, 113, 108, 6, 15, 2, 41, 59, 10, 4”
The extended Mittag-Leffler function via fractional calculus (2017, background): “Some interesting applications of this function are considered in the study of kinetic equation, the telegraph equations, random walks, Levy flights, supper diffuse transport, complex system, non-equilibrium statistical mechanics and quantum mechanics, interested readers can refer [2, 4, 5, 11, 16, 2”
Explicit formulas for the Dunkl dihedral kernel and the $(\kappa, a)$-generalized Fourier kernel (2016, methodology): “i+ −i 2 . It is possible to give an integral expression of this kernel in terms of the generalized Mittag-Leﬄer function. We give the deﬁnition and its Laplace transform here, see also Chapter 2 in [22]. 14 Deﬁnition 1. The generalized Mittag-Leﬄer function is deﬁned by Eδ ǫ,γ(z) := X∞ n=0 (δ)nzn Γ(”
The Role of the Mittag-Leffler Function in Fractional Modeling (2015, background): “Asymptotic properties of the Mittag-Leffler-type functions show their place in the whole theory of special functions (see also [30]).”
Multi-parametric mittag-leffler functions and their extension (2013, background): “Solutions of fractional differential equations by the operational methods are also obtained in terms of certain multivariate Mittag-Leffler functions of n complex variables z1, . . . , zn, defined as (see, e.g., Luchko [37], KilbasSrivastava-Trujillo [30], Haubold-Mathai-Saxena [18], etc.): Ea1,...,”
Special matrix functions: characteristics, achievements and future directions (2018, background): “Also, recently special matrix functions appear in connection with statistics [8–10], Lie groups theory [11], mathematical physics and theoretical physics (see, [12,13]).”
Perfect hedging in rough Heston models (2017, unknown): “The following properties of fα,λ and Fα,λ can be found in [14, 17, 18].”
The Chronicles of Fractional Calculus (2017, unknown): “1980-1989 [183, 184, 146, 136, 162, 176, 137, 138] [125] 1990 [186] [68] 1991 [140, 139, 147, 61] 1992 [182, 33, 177, 130] [80] 1993 1994 [206, 85] 1995 [148] [172] 1996 [141, 170] 1997 [34, 118] 1998 [128] [173, 123] 1999 [158, 150, 193, 199] 2000 [67] 2001 [9, 13] [16] 2002 [201, 157, 54, 73, 72] ”