Related papers: Fractional moments

Fractional vector calculus is discussed in the spherical coordinate framework. A variation of the Legendre equation and fractional Bessel equation are solved by series expansion and numerically. Finally, we generalize the hypergeometric…

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.

We consider and provide an accurate study for the fractional Zernike functions on the punctured unit disc, generalizing the classical Zernike polynomials and their associated $\beta$-restricted Zernike functions. Mainly, we give the…

This work, dealt with the classical mean value theorem and took advantage of it in the fractional calculus. The concept of a fractional critical point is introduced. Some sufficient conditions for the existence of a critical point is…

Fractional derivative in time variable is introduced into the Fokker-Planck equation of a population growth model. It's solution, the KNO scaling function, is transformed into the generating function for the multiplicity distribution.…

We consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples. The corresponding equations of motion are derived using variational principle.

By using definition of Golden derivative, corresponding Golden exponential function and Fibonomial coefficients, we introduce generating functions for Bernoulli-Fibonacci polynomials and related numbers. Properties of these polynomials and…

Several second moment and other integral evaluations for the Riemann zeta function $\zeta(s)$, Hurwitz zeta function $\zeta(s,a)$, and Lerch zeta function $\Phi(z,s,a)$ are presented. Additional corollaries that are obtained include…

We investigate fractional moments and expectations of power means of complex-valued random variables by using fractional calculus. We deal with both negative and positive orders of the fractional derivatives. The one-dimensional…

We consider the set of power functions defined on the set of positive real number, and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative…

Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and…

Integral expressions for positive-part moments E X_+^p (p>0) of random variables X are presented, in terms of the Fourier-Laplace or Fourier transforms of the distribution of X. A necessary and sufficient condition for the validity of such…

We give a short review of recent progress on determining the order of magnitude of moments $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ of random multiplicative functions, and of closely related issues. We hope this can serve as a concise…

We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…

Let $(F_n)_{n\ge 1}$ be the Fibonacci sequence. Define $P(F_n): = (\sum_{i=1}^n F_i)_{n\ge 1}$; that is, the function $P$ gives the sequence of partial sums of $(F_n)$. In this paper, we first give an identity involving $P^k(F_n)$, which is…

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canonical Hamiltonian are given, and a set of…

Let $\{b_H(t),t\in\mathbb{R}\}$ be the fractional Brownian motion with parameter $0<H<1$. When $1/2<H$, we consider diffusion equations of the type \[X(t)=c+\int_0^t\sigma\bigl(X(u)\bigr)\mathrm {d}b_H(u)+\int _0^t\mu\bigl(X(u)\bigr)\mathrm…

Here we introduce a fractional notion of $k$-dimensional measure, $0\leq k<n$, that depends on a parameter $\sigma$ that lies between $0$ and $1$. When $k=n-1$ this coincides with the fractional notions of area and perimeter, and when $k=1$…

Singular functions and, in general, H\"older functions represent conceptual models of nonlinear physical phenomena. The purpose of this survey is to demonstrate the applicability of fractional velocity as a tool to characterize Holder and…

This paper gives a brief introduction to some important fractional and multifractional Gaussian processes commonly used in modelling natural phenomena and man-made systems. The processes include fractional Brownian motion (both standard and…