Some Fractional Special Functions and Fractional Moments (original) (raw)
Generalized Fourier Transform for the Generation of Complex Fractional Moments
2014
F ourier transform of fractional order using the Mittag-Leffler-type function and its complex type, was introduced together with its inversion formula. The obtained transform provided a suitable generalization of the characteristic function of random variables. It was shown that complex fractional moments which are complex moments of order nq th of a certain distribution, are equivalent to Caputa fractional derivation of generalized characteristic function (GCF) in origin, n being a positive integer and 0 < q ≤ 1 . The case q=1 was reduced to the complex moments. Finally, after introducing fractional factorial moments of a positive random variable, we presented the relationship between integer moments, fractional moments (FMs) and fractional factorial moments (FFMs) of a positive random variable.
Stieltjes moment problem via fractional moments
Applied Mathematics and Computation, 2005
Stieltjes moment problem is considered to recover a probability density function from the knowledge of its infinite sequence of ordinary moments. The approximate density is obtained through maximum entropy technique, under the constraint of few fractional moments. The latter are numerically obtained from the infinite sequence of ordinary moments and are chosen in such a way as to convey the maximum information content carried by the ordinary moments. As a consequence a model with few parameters is obtained and intrinsic numerical instability is avoided. It is proved that the approximate density is useful for calculating expected values and some other interesting probabilistic quantities.
Advances in Difference Equations, 2016
We aim to investigate the MSM-fractional calculus operators, Caputo-type MSM-fractional differential operator, and pathway fractional integral operator of the generalized k-Mittag-Leffler function. We also investigate certain statistical distribution associated with the generalized k-Mittag-Leffler function. Certain particular cases of the derived results are considered and indicated to further reduce to some known results.
International Journal of Advanced Statistics and Probability, 2013
This paper derives a probability distribution with fractional moments arising from generalized Pearson system of differential equation. The expressions for the probability density function, cumulative distribution function and moments have been obtained. The plots for the probability density function, cumulative distribution function, and survival and hazard functions are also provided. Some distributional relationships of the proposed distribution have been established. A characterization of the new distribution is given. It is hoped that the findings of the paper will be useful for researchers in different fields like economics, engineering, environmental science, finance, medical sciences and physical sciences among others, where fractional moments become required to be computed if the integer moments 1 k do not exist.
A generalized moments expansion
Physics Letters A, 2006
A new generalized moments expansion (GMX), which is based on the t-expansion, is derived. The well-known connected moments expansion (CMX) and alternate moments expansion (AMX) are shown to be special cases of the GMX.
An Application of Discrete Fractional Calculus in Statistics
2018
We introduced generalized random variables of discrete type, studied some of their properties and then related these to continuous random variable which has been studied by Ganji and Gharari [8]. With this introduction, we obtained a new relationship between discrete fractional calculus and statistics. Also, the fractional versions of the discrete uniform distribution are developed and their statistical properties are discussed. KEY WORDS: Fractional sum, Fractional difference, Discrete uniform distribution, Generalized random variable. MSC: 60E05, 39A12.
An Introduction to the Generalized Fractional Integration
Boletim da Sociedade …
Abstract: The purpose of the present paper is to investigate the generalizedfractional integration of the generalized M-series. Some results derived by Saxena and Saigo [13], Samko, Kilbas and Marichev [15] are the special cases of the main results derived in this ...
Hausdorff moment problem via fractional moments
Applied Mathematics and Computation, 2003
We outline an efficient method for the reconstruction of a probability density function from the knowledge of its infinite sequence of ordinary moments. The approximate density is obtained resorting to maximum entropy technique, under the constraint of some fractional moments. The latter ones are obtained explicitly in terms of the infinite sequence of given ordinary moments. It is proved that the approximate density converges in entropy to the underlying density, so that it demonstrates to be useful for calculating expected values.
Hausdorff moment problem and fractional moments: A simplified procedure
Applied Mathematics and Computation, 2011
Hausdorff moment problem is considered and a solution, consisting of the use of fractional moments, is proposed. More precisely, in this work a stable algorithm to obtain centered moments from integer moments is found. The algorithm transforms a direct method into an iterative Jacobi method which converges in a finite number of steps, as the iteration Jacobi matrix has null spectral radius. The centered moments are needed to calculate fractional moments from integer moments. As an application few fractional moments are used to solve finite Hausdorff moment problem via maximum entropy technique. Fractional moments represent a remedy to ill-conditioning coming from an high number of integer moments involved in recovering procedure.
Fractional Calculus and Special Functions 1 Fractional Calculus and Special Functions
2008
The aim of these introductory lectures is to provide the reader with the essentials of the fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes. We discuss the linear operators of fractional integration and fractional differentiation, which were introduced in pioneering works by Abel, Liouville, Riemann, Weyl, Marchaud, M. Riesz, Feller and Caputo. Particular attention is devoted to the techniques of Fourier and Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. Furthermore, we discuss the approach based on limit of difference quotients, formerly introduced by Grünwald and Letnikov, which provides a discrete access to the fractional calculus. Such approach is very useful for actual numerical computation and is complementary to the previous integral approaches, which provide the continuous access to the fractional calculus. Finally, we give some information on the higher transcendental functions of the Mittag-Leffler and Wright type which, together with the most common Eulerian functions, turn out to play a fundamental role in the theory and applications of the fractional calculus. We refrain for treating the more general functions of the Fox type (H functions), referring the interested reader to specialized papers and books.