Related papers: A Generalized Representation of Fa\'a di Bruno's F…
In this manuscript, we generalize F-calculus to apply it on fractal Tartan spaces. The generalized standard F-calculus is used to obtain the integral and derivative of the functions on the fractal Tartan with different dimensions. The…
We study the global fluctuations for linear statistics of the form $\sum_{i=1}^n f(\lambda_i)$ as $n \rightarrow \infty$, for $C^1$ functions $f$, and $\lambda_1, ..., \lambda_n$ being the eigenvalues of a (general) $\beta$-Jacobi ensemble,…
In analogy to the definition of the lambda-determinant, we define a one-parameter deformation of the Dodgson condensation formula for Pfaffians. We prove that the resulting rational function is a polynomial with weights given by the…
Fractional diffusion and Fokker-Planck equations are widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be…
In this paper, we introduce a family of robust estimates for the parametric and nonparametric components under a generalized partially linear model, where the data are modeled by $y_i|(\mathbf{x}_i,t_i)\sim F(\cdot,\mu_i)$ with…
In this paper we deal with Mellin convolution of generalized Gamma densities which leads to integrals of modified Bessel functions of the second kind. Such convolutions allow us to explicitly write the solutions of the time-fractional…
By establishing an interesting connection between ordinary Bell polynomials and rational convolution powers, some composition and inverse relations of Bell polynomials as well as explicit expressions for convolution roots of sequences are…
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…
For any function F(x) having a Taylor expansion we solve the boson normal ordering problem for F[(a*)^r a^s], with r,s positive integers,[a,a*]=1, i.e. we provide exact and explicit expressions for its normal form which has all a's to the…
The aim of this paper is to represent any polynomial in terms of the degenerate Frobenius-Euler polynomials and more generally of the higher-order degenerate Frobenius-Euler polynomials. We derive explicit formulas with the help of umbral…
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobinski…
We provide an introduction of some basic facts of uniformly almost periodic functions, such as Fourier series representations. A result is then proved about Fourier coefficients which is a generalization of the purely periodic case. We then…
Consider a normal function $f$ on the ordinals (i. e. a function $f$ that is strictly increasing and continuous at limit stages). By enumerating the fixed points of $f$ we obtain a faster normal function $f'$, called the derivative of $f$.…
The concept of a composed product for univariate polynomials has been explored extensively by Brawley, Brown, Carlitz, Gao, Mills, et al. Starting with these fundamental ideas and utilizing fractional power series representation (in…
We prove a Faa di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids.
In this paper we show several connections between special functions arising from generalized COM-Poisson-type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New…
Given two polynomials f(x) and g(x), we extend the formula expressing the remainder in terms of the roots of these two polynomials to the case where f(x) is a Laurent polynomial. This allows us to give new expressions of a Schur function,…
In the present article, a new method for the evaluation of fractional derivatives of arbitrary real order is proposed. Numerous but inequivalent formulations have been given in the past. Some of them exhibit unsatisfactory properties such…
We discuss certain aspects of the formal calculus used to describe vertex algebras. In the standard literature on formal calculus, the expression $(x+y)^{n}$, where $n$ is not necessarily a nonnegative integer, is defined as the formal…
The fractional Leibniz rule is generalized by the Coifman-Meyer estimate. It is shown that the arbitrary redistribution of fractional derivatives for higher order with the corresponding correction terms.