Related papers: A short proof of solution formulas for the linear …
We give a short proof, using generating functions, for a polynomial congruence for Eulerian polynomials first proved, using arrangements of hyperplanes, by Yoshinaga and later proved, using roots of unity, by Iijima, Sasaki, Takahashi, and…
In this work, we describe our experience in learning the use of a computer proof assistant - specifically, Lean - from scratch, through proving formulae for the solutions of polynomial equations. Specifically, in this work we characterize…
Cubic and quartic non-autonomous differential equations with continuous piecewise linear coefficients are considered. The main concern is to find the maximum possible multiplicity of periodic solutions. For many classes, we show that the…
We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…
The well-known mathematical instrument for detection common roots for pairs of polynomials and multiple roots of polynomials are resultants and discriminants. For a pair of polynomials $f$ and $g$ their resultant $R(f,g)$ is a function of…
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…
The roots of a complex polynomial depend continuously on the coefficients; that is, an infinitesimal perturbation of the coefficients results in an infinitesimal perturbation of the roots. A short, straightforward proof of this is possible…
We show that Pinney's equation [2] with a constant coefficient can be reduced to its linear part by a simple change of variables. Also, Pinney's original solution is simplified slightly.
The method of this paper is my original creation. A new method for solving linear differential equations is proposed in this paper. The important conclusion of this paper is that arbitrary order linear ordinary differential equations with…
In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…
Univariate polynomials with only real roots -- while special -- do occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of two young mathematicians, Julius Borcea and…
This paper presents new formulary solutions for quantic polynomial equations in general forms, where we present five solutions for any fifth degree polynomial equation with real coefficients, and thereby having the possibility to calculate…
The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.
This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to…
In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…
This work deals with the existence of an almost periodic solution for certain kind of differential equations with generalized piecewise constant argument, almost periodic coefficients which are seen as a perturbation of a linear equation of…
We revisit the classical problem of construction of a fundamental system of solutions to a linear ODE whose elements remain analytic and linearly independent for all values of the roots of the characteristic polynomial.
In this work we consider a given root of a family of n-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). More…
Translation of the Latin original, "Methodus generalis investigandi radices omnium aequationum per approximationem" (1776). E643 in the Enestrom index. Euler gives a series to find powers of roots of polynomials.
Recently new solvable systems of nonlinear evolution equations -- including ODEs, PDEs and systems with discrete time -- have been introduced. These findings are based on certain convenient formulas expressing the $k$-th time-derivative of…