Characterizing Polynomials By Forward Differences

Journal of Approximation Theory, 1990

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Journal of The London Mathematical Society-second Series, 1983

We show here, among other results, that there exist two monic polynomials P,Q with integral coefficients which have the same set of complex roots and are such that their derivatives also have the same roots, but there are no positive integers m, n such that P m = Q".

2005

We generalize Flett’s Mean Value Theorem to the case of functions defined in normed spaces. This is a motivation for considering functional equations related to the Flett mean value formula in a quite general setting. We solve them in the case where functions are defined in abelian groups and take values in a rational linear space.

International Journal of Nonlinear Analysis and Applications, 2015

For every $1leq s< n$, the $s^{th}$ derivative of a polynomial $P(z)$ of degree $n$ is a polynomial $P^{(s)}(z)$ whose degree is $(n-s)$. This paper presents a result which gives generalizations of some inequalities regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle. Besides, our result gives interesting refinements of some well-known results.

Bulletin of the London Mathematical Society, 1985

Iranian Journal of Mathematical Sciences and Informatics, 2017

The paper presents an L r − analogue of an inequality regarding the s th derivative of a polynomial having zeros outside a circle of arbitrary radius but greater or equal to one. Our result provides improvements and generalizations of some well-known polynomial inequalities.

arXiv (Cornell University), 2009

We show that the constant $\frac{1}{cos^2(\frac{\pi}{n+2})}$ in this theorem could not be increased. We also present some corollaries of this theorem.

Journal of Mathematical Analysis and Applications, 2008

A real polynomial is called Hurwitz (stable) if all its zeros have negative real parts. For a given n ∈ N we find the smallest possible constant d n > 0 such that if the coefficients of F (z) = a 0 + a 1 z + • • • + a n z n are positive and satisfy the inequalities a k a k+1 > d n a k−1 a k+2 for k = 1, 2,. .. ,n − 2, then F (z) is Hurwitz.

Aequationes mathematicae, 2005

This paper is about the characterization of those functions which can be expressed as a sum of a generalized polynomial and a generalized logarithmic polynomial. The simplest case of generalized polynomials and generalized logarithmic polynomials of the first degree is to characterize those functions which are sums of affine and logarithmic functions. This problem is solved in . The main result of this article is the following characterization theorem: A function f : R + → R is the sum of a generalized polynomial of degree at most n and of a generalized logarithmic polynomial of degree at most m if and only if all of its m-th multiplicative differences are generalized polynomials of degree at most n.

For an entire function f , we define the difference operators as

2009

It is shown that if w(z) is a finite-order meromorphic solution of the equation H(z,w) P(z,w) = Q(z,w), where P(z,w) = P(z,w(z),w(z+c_1),...,w(z+c_n)) is a homogeneous difference polynomial with meromorphic coefficients, and H(z,w) = H(z,w(z)) and Q(z,w) = Q(z,w(z)) are polynomials in w(z) with meromorphic coefficients having no common factors such that max{deg_w(H), deg_w(Q) - deg_w(P)} > min{deg_w(P), ord_0(Q)} - ord_0(P), where ord_0(P) denotes the order of zero of P(z,x_0,x_1,...,x_n) at x_0=0 with respect to the variable x_0, then the Nevanlinna counting function N(r,w) satisfies N(r,w) > S(r,w). This implies that w(z) has a relatively large number of poles. For a smaller class of equations a stronger assertion N(r,w) = T(r,w)+S(r,w) is obtained, which means that the pole density of w(z) is essentially as high as the growth of w(z) allows. As an application, a simple necessary and sufficient condition is given in terms of the value distribution pattern of the solution, which can be used as a tool in ruling out the possible existence of special finite-order Riccati solutions within a large class of difference equations containing several known difference equations considered to be of Painleve type.

We prove that the question of whether a given linear partial differential or difference equation with polynomial coefficients has non-zero polynomial solutions is algorithmically undecidable. However, for equations with constant coefficients this question can be decided very easily since such an equation has a non-zero polynomial solution iff its constant term is zero. We give a simple combinatorial proof of the fact that in this case the equation has polynomial solutions of all degrees. For linear partial q-difference equations with polynomial coefficients, the question of decidability of existence of non-zero polynomial solutions remains open. Nevertheless, for such equations with constant coefficients we show that the space of polynomial solutions can be described algorithmically. We present examples which demonstrate that, in contrast with the differential and difference cases where the dimension of this space is either infinite or zero, in the q-difference case it can also be finite and non-zero.

Functiones et Approximatio Commentarii Mathematici

In this paper, we study the derivatives of an integer-valued polynomial of a given degree. Denoting by E n the set of the integer-valued polynomials with degree ≤ n, we show that the smallest positive integer c n satisfying the property: ∀P ∈ E n , c n P ′ ∈ E n is c n = lcm(1, 2,. .. , n). As an application, we deduce an easy proof of the well-known inequality lcm(1, 2,. .. , n) ≥ 2 n−1 (∀n ≥ 1). In the second part of the paper, we generalize our result for the derivative of a given order k and then we give two divisibility properties for the obtained numbers c n,k (generalizing the c n 's). Leaning on this study, we conclude the paper by determining, for a given natural number n, the smallest positive integer λ n satisfying the property: ∀P ∈ E n , ∀k ∈ N: λ n P (k) ∈ E n. In particular, we show that: λ n = p prime p ⌊ n p ⌋ (∀n ∈ N).

LIMITED POLYNOMIALS, 2021

In this paper we study a particular class of polynomials. We study the distribution of their zeros, including the zeros of their derivatives as well as the interaction between this two. We prove a weak variant of the sendov conjecture in the case the zeros are real and are of the same sign.