Related papers: Closed-form formula for some recursively-defined i…
In the world of mathematical analysis, many counterintuitive answers arise from the manipulation of seemingly unrelated concepts, ideas, or functions. For example, Euler showed that $e^{i\pi} + 1 = 0$, whereas Gauss proved that the area…
We prove a general statement about the integrality of the sequences generated by a recursion of the following form: $nu_n$ equals a linear combination of $u_{n-1},u_{n-2},\dots,u_0$ with polynomial coefficients in $n$ of special form. This…
For the first time, the general nonlinear Schr\"odinger equation is investigated, in which the chromatic dispersion and potential are specified by two arbitrary functions. The equation in question is a natural generalization of a wide class…
The present study introduces and investigates a new type of equation which is called Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves…
The Riemann zeta function can be written as the Mellin transform of the unit interval map w(x) = floor(1/x)*(-1+x*floor(1/x)+x) multiplied by s((s+1)/(s-1)). A finite-sum approximation to \zeta (s) denoted by \zeta_w(N;s) which has real…
The definite integrals $ \int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^3\D x$, $ \int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^2P_{\nu}(-x)\D x$, $\int_{-1}^1x(1-x^2)^{(\nu-1)/2}[P_{\nu+1}(x)]^3\D x $ and…
A unified explicit form for difference formulas to approximate the fractional and classical derivatives is presented. The formula gives finite difference approximations for any classical derivatives with a desired order of accuracy at nodal…
The convolution of a function with an isotropic Gaussian appears in many contexts such as differential equations, computer vision, signal processing, and numerical optimization. Although this convolution does not always have a closed form…
Integral Cauchy theorem is used to derive closed-form expressions of the roots of a univariate polynomial of any degree as integrals of elementary functions.
We study differential-difference equation of the form $t_{x}(n+1)=f(t(n),t(n+1),t_x(n))$ with unknown $t=t(n,x)$ depending on $x$, $n$. The equation is called Darboux integrable, if there exist functions $F$ (called an $x$-integral) and $I$…
We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo- and photo-…
We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the behavior of a solution whose derivatives are…
Our goal in this work is to found a closed form for rational generat- ing functions, these generate a various families of polynomials and generalized polynomials, in order to get the general recursive formula satisfied by these polynomials.
In the present paper, two discrete forms of Appell function $F_1$ are introduced and studied. We examine the first discrete form in detail and give the results directly for the second. We study their regions of convergence, differential…
Euler discovered a formula for expressing the value of the Riemann zeta function for all even positive integer arguments. A closed-form expression for the Riemann zeta function for all odd integer arguments, based on the values of the…
The integral $\int_{-\infty}^{\infty} e^{- x^2 - g x^4} dx $ is used as an introductory learning tool in the study of Quantum Field Theory and path integrals. Typically it is analysed via perturbation theory. Close form solutions have been…
This paper introduces the notion of referring forms as a new metric for analyzing sequential circuits from a functional perspective. Sequential circuits are modeled as causal stream functions, the outputs of which depend solely on the past…
We derive a concise closed-form solution for a linear three-term recurrence relation. Such recurrence relations are very common in the quantitative sciences, and describe finite difference schemes, solutions to problems in Markov processes…
Let $\spt(n)$ be the number of smallest parts in the partitions of $n$. In this paper, we give some generalized Euler-like recursive formulas for the $\spt$ function in terms of Hecke trace of values of special twisted quadratic Dirichlet…
Our purpose in this paper is to prove, under some regularity conditions on the datas, the solvability in a Gevrey class of bound -1 on the interval [-1,1] of a class of nonlinear fractional functional differential equations.