Related papers: Higher order Poincare-Pontryagin functions and ite…
The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\pi(x)$ having a general expression of the form, \begin{align*} P(\pi(x)) - \frac{e x}{\log x} Q(\pi(x/e)) +…
We study the higher order q- Poincar\'e and other coercive inequalities for a class probability measures satisfying Adam's regularity condition.
In this paper we combine the fractional $\psi-$hyperholomorphic function theory with the fractional calculus with respect to another function. As a main result, a fractional Borel-Pompeiu type formula related to a fractional $\psi-$Fueter…
We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented…
The normal function associated to algebraic cycles in higher Chow groups defines a differential equation. This Picard-Fuchs equation defines an extension of $D$-modules as well as an extension of local systems. In this paper, we show that…
We give new bounds on the zeros of polynomials in $z$ and the $j$--function, and $z$ and Weierstrass elliptic functions with rectangular associated lattice, controlling the zeros of these functions by their tame behaviour on the boundaries…
We analyze the iterative structure of unfactorized partonic structure functions in the large-x limit, and derive all-order expressions for the leading-logarithmic off-diagonal splitting functions P_gq and P_qg and the corresponding…
Following the ideas of L. Carlitz we introduce a generalization of the Bernoulli and Eulerian polynomials of higher order to vectorial index and argument. These polynomials are used for computation of the vector partition function $W({\bf…
Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of Brouwer degree theory. Between the…
In this article, we prove an asymptotic formula for mean values of long Dirichlet polynomials with higher order shifted divisor functions, assuming a smoothed additive divisor conjecture for higher order shifted divisor functions. As a…
Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the…
We present an approach for construction of functional bases of differential invariants for some infinite-dimensional algebras with coefficients of generating operators depending on arbitrary functions. An example for the…
We address the Poincar\'e-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in…
D. Leviatan has investigated the behavior of the higher order derivatives of approximation polynomials of the differentiable function $f$ on $[-1,1]$. Especially, when $P_n$ is the best approximation of $f$, he estimates the differences…
In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the form $$g(z,\varepsilon)=g_0(z)+\sum_{i=1}^k \varepsilon^i g_i(z)+\mathcal{O}(\varepsilon^{k+1}),$$ for $|\varepsilon|\neq0$…
Using a simple transfer matrix approach we have derived very long series expansions for the perimeter generating function of punctured staircase polygons (staircase polygons with a single internal staircase hole). We find that all the terms…
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…
The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of…
It is known that the Principal Poincar\'e Pontryagin Function is generically an Abelian integral. In non generic cases it is an iterated integral. In previous papers one of the authors gives a precise description of the Principal Poincar\'e…
The Dyson-Schwinger (DS) equations for a quantum field theory in $D$-dimensional space-time are an infinite sequence of coupled integro-differential equations that are satisfied exactly by the Green's functions of the field theory. This…