Related papers: On Polynomial Solutions of Linear Differential Equ…
Given an ordinary differential field $K$ of characteristic zero, it is known that if $y$ and $1/y$ satisfy linear differential equations with coefficients in $K$, then $y'/y$ is algebraic over $K$. We present a new short proof of this fact…
In a classical case, orthogonal polynomial sequences are in such a way that the $ n $th polynomial has the exact degree $n$. Such sequences are complete and form a basis of the space for any arbitrary polynomial. In this paper, we introduce…
For a sequence of polynomials $\{p_k(t)\}$ in one real or complex variable, where $p_k$ has degree $k$, for $k\ge 0$, we find explicit expressions and recurrence relations for infinite matrices whose entries are the coefficients $d(n,m,k)$,…
In this paper we study the existence of continuous solutions and their constructions for a second order iterative functional equation, which involves iterate of the unknown function and a nonlinear term. Imposing Lipschitz conditions to…
In this paper, we study the transcendental meromorphic solutions for the nonlinear differential equations: $f^{n}+P(f)=R(z)e^{\alpha(z)}$ and $f^{n}+P_{*}(f)=p_{1}(z)e^{\alpha_{1}(z)}+p_{2}(z)e^{\alpha_{2}(z)}$ in the complex plane, where…
We propose a method to compute the numerical solutions of a polynomial system in complete intersection. This algorithm makes use of Bezout matrices and need only linear algebra computations. All the calculations can be done in floating…
We show that for all homogeneous polynomials $ f_{m}$ of degree $m$, in $d$ variables, and each $j = 1, \dots , d$, we have \begin{equation*} \left\langle x_{j}^{2}f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb{S}% ^{d-1}\right) } \geq…
Renormalized homotopy continuation on toric varieties is introduced as a tool for solving sparse systems of polynomial equations, or sparse systems of exponential sums. The cost of continuation depends on a renormalized condition length,…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially…
In this paper, we provide a necessary and sufficient condition ensuring the property of exponential dichotomy for periodic linear systems of generalized differential equations. This condition allow us to revisit a recent result of…
In the paper, we introduce a matrix method to constructively determine spaces of polynomial solutions (in general, multiplied by exponentials) to a system of constant coefficient linear PDE's with polynomial (multiplied by exponentials)…
Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equation. We apply this principle by finding dilatations and…
We resolve the Ramsey problem for $\{x,y,z:x+y=p(z)\}$ for all polynomials $p$ over $\mathbb{Z}$. In particular, we characterise all polynomials that are $2$-Ramsey, that is, those $p(z)$ such that any $2$-colouring of $\mathbb{N}$ contains…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
This paper presents a universal numerical scheme tailored for tackling linear integral, integro-differential, and both initial and boundary value problems of ordinary differential equations. The numerical scheme is readily adapted for…
The paper represents the method for construction of the families of particular solutions to some new classes of $(n+1)$ dimensional nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
We continue to investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation. In an earlier article we had introduced the distinction between periodic and…
In this paper, we will give suitable conditions on differential polynomials $Q(f)$ such that they take every finite non-zero value infinitely often, where $f$ is a meromorphic function in complex plane. These results are related to Problem…