Some Fractional Special Functions and Fractional Moments (original) (raw)
Related papers
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.
Mathematics, 2019
Several fractional calculus operators have been introduced and investigated. In this sequence, we aim to establish the Marichev-Saigo-Maeda (MSM) fractional calculus operators and Caputo-type MSM fractional differential operators of extended Mittag-Leffler function (EMLF). We also investigate the statistical distribution associated with the EMLF. Finally, we derive some of the particular cases of the main results.
Generalized Moment Generating Functions of Random Variables and Their Probability Density Functions
This paper seeks to develop a generalized method of generating the moments of random variables and their probability distributions. The Generalized Moment Generating Function is developed from the existing theory of moment generating function as the expected value of powers of the exponential constant. The methods were illustrated with the Beta and Gamma Family of Distributions and the Normal Distribution. The methods were found to be able to generate moments of powers of random variables enabling the generation of moments of not only integer powers but also real positive and negative powers. Unlike the traditional moment generating function, the generalized moment generating function has the ability to generate central moments and always exists for all continuous distribution but has not been developed for any discrete distribution. Cite This Article: Matthew Chukwuma Michael, Oyeka Cyprain Anene, and Ashinze Mpuruoma Akudo, " Generalized Moment Generating Functions of Random Variables and Their Probability Density Functions. "
An extension of beta function, its statistical distribution, and associated fractional operator
Advances in Difference Equations, 2020
Recently, various forms of extended beta function have been proposed and presented by many researchers. The principal goal of this paper is to present another expansion of beta function using Appell series and Lauricella function and examine various properties like integral representation and summation formula. Statistical distribution for the above extension of beta function has been defined, and the mean, variance, moment generating function and cumulative distribution function have been obtained. Using the newly defined extension of beta function, we build up the extension of hypergeometric and confluent hypergeometric functions and discuss their integral representations and differentiation formulas. Further, we define a new extension of Riemann–Liouville fractional operator using Appell series and Lauricella function and derive its various properties using the new extension of beta function.
A New Generalized Moment Generating Function of Random Variables
Applied Mathematics & Information Sciences
Many of the important characteristics and features of a distribution are obtained through the ordinary moments and generating function. The main goal of this paper is to address a new approach to compute, without using multiple integrals and derivatives, E (X a +b) r (X c +d) s for a nonnegative random variable, where a, b, c, d are any real number. The proposed approach is discussed in detail and illustrated through a few examples.
Some applications of the fractional Poisson probability distribution
Journal of Mathematical Physics, 2009
Physical and mathematical applications of fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states has been introduced and studied. As mathematical applications, we have discovered and developed the fractional generalization of Bell polynomials, Bell numbers, and Stirling numbers. Appearance of fractional Bell polynomials is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states. Fractional Stirling numbers of the second kind have been applied to evaluate skewness and kurtosis of the fractional Poisson probability distribution function. A new representation of Bernoulli numbers in terms of fractional Stirling numbers of the second kind has been obtained. A representation of Schläfli polynomials in terms of fractional Stirling numbers of the second kind has been found. A new representations of Mittag-Leffler function involving fractional Bell polynomials and fractional Stirling numbers of the second kind have been discovered. Fractional Stirling numbers of the first kind have been introduced and studied. Two new polynomial sequences associated with fractional Poisson probability distribution have been launched and explored. The relationship between new polynomials and the orthogonal Charlier polynomials has also been investigated. * In the limit case when the fractional Poisson probability distribution becomes the Poisson probability distribution, all of the above listed developments and implementations turn into the well-known results of quantum optics, the theory of combinatorial numbers and the theory of orthogonal polynomials of discrete variable.