Linear Differential Polynomials Research Articles

On the uniqueness problems of entire functions and their linear differential polynomials

On the uniqueness problems of entire functions and their linear differential polynomials

Entire function with its linear differential polynomial normally or weakly shares a set and some related topics

Entire function with its linear differential polynomial normally or weakly shares a set and some related topics

An Entire Function that Shares a Small Function with its Derivative and Linear Differential Polynomial

An Entire Function that Shares a Small Function with its Derivative and Linear Differential Polynomial

Une généralisation du critère de Boulier–Buchberger pour le calcul des ensembles caractéristiques d’idéaux différentiels

We generalize the analog of Buchberger's first criterion, stated by Boulier et al., for detecting useless S-polynomials reductions in the computation of characteristic sets of differential ideals. The original version assumes linear polynomials; this result is here extended to a product of linear differential polynomials depending of the same arbitrary differential polynomial.

Power of Entire Function Sharing Non-zero Polynomials with It’s Linear Differential Polynomial

Power of Entire Function Sharing Non-zero Polynomials with It’s Linear Differential Polynomial

LINEAR DIFFERENTIAL POLYNOMIALS WEIGHTED-SHARING A SET OF ROOTS OF UNITY

In this paper, we study the uniqueness of linear diFFerential polynomials of meromorphic functions when they share a set of roots of unity. Our results shall generalize recent results.

Growth of solutions of non-homogeneous linear differential equations and its applications

In this paper, we investigate the growth properties of solutions of the non-homogeneous linear complex differential equation $L\left(f\right)=b\left(z\right)f+c\left(z\right)$, where $L\left(f\right)$ is a linear differential polynomial and $b\left( z\right) $, $c\left( z\right) $ are entire functions and give some of its applications on sharing value problems.

Value sharing of non linear differential polynomials with the glimpse of normal family

Value sharing of non linear differential polynomials with the glimpse of normal family

An Inverse Problem for LTV ODE’s

It is well known that any linear combination of products of polynomials and exponentials (a.k.a. Bohl functions) are annihilated by LTI-ODE’s. In this paper, we consider the more general classes of piecewise Bohl functions and of real analytic functions and present the realization of the latter as solutions of linear differential polynomials with real analytic coefficients. In particular, we show that the minimal order depends solely on the multiplicity of the zeros of the given function. We extend the problem further to meromorphic functions without poles on the real line. We briefly comment on piecewise-defined functions and the nonhomogeneous problem.

Tropical Differential Gröbner Bases

In this paper, the concept of tropical differential Grobner basis is introduced, which is a natural generalization of the tropical Grobner basis to the recently introduced tropical differential algebra. Like the differential Grobner basis, the tropical differential Grobner basis generally contains an infinite number of elements. We give a Buchberger style criterion for the tropical differential Grobner basis. For differential ideals generated by homogeneous linear differential polynomials with constant coefficients, we give a complete algorithm to compute the finite tropical differential Grobner basis.

LINEAR DIFFERENTIAL POLYNOMIALS SHARING A SET OF THE ROOTS OF UNITY

LINEAR DIFFERENTIAL POLYNOMIALS SHARING A SET OF THE ROOTS OF UNITY

Uniqueness of meromorphic functions of differential polynomials sharing a small function with finite weight

Abstract Let f be a non-constant meromorphic function and {a=a(z)} ( {\not\equiv 0,\infty} ) a small function of f. Here, we obtain results similar to the results due to Indrajit Lahiri and Bipul Pal [Uniqueness of meromorphic functions with their homogeneous and linear differential polynomials sharing a small function, Bull. Korean Math. Soc. 54 2017, 3, 825–838] for a more general differential polynomial by introducing the concept of weighted sharing.

Results on entire and meromorphic functions that share small function with their homogeneous and linear differential polynomials

Using the results of S. S. Bhoosnurmath, we mainly study the uniqueness of entire and meromorphic functions that share small functions with their homogeneous and linear differential polynomials. In this paper, we obtain significant improvements and generalizations of the results of H. X. Yi.

Computing Approximate Greatest Common Right Divisors of Differential Polynomials

Differential (Ore) type polynomials with “approximate” polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate greatest common right divisor problem (GCRD) of differential polynomials, as a non-commutative generalization of the well-studied approximate GCD problem. Given two differential polynomials, we present an algorithm to find nearby differential polynomials with a non-trivial GCRD, where nearby is defined with respect to a suitable coefficient norm. Intuitively, given two linear differential polynomials as input, the (approximate) GCRD problem corresponds to finding the (approximate) differential polynomial whose solution space is the intersection of the solution spaces of the two inputs. The approximate GCRD problem is proven to be locally well posed. A method based on the singular value decomposition of a differential Sylvester matrix is developed to produce an initial approximation of the GCRD. With a sufficiently good initial approximation, Newton iteration is shown to converge quadratically to an optimal solution. Finally, sufficient conditions for existence of a solution to the global problem are presented along with examples demonstrating that no solution exists when these conditions are not satisfied.

Meromorphic Functions Sharing Three Values with Their Linear Differential Polynomials in an Angular Domain

Let f be a nonconstant meromorphic function of lower order µ (f) > 1/2 in ℂ, and let aj (j = 1, 2, 3) be three distinct finite complex numbers. We show that there exists an angular domain D = {z: α ≤ arg z ≤ β}, where 0 < β − α ≤ 2π, such that if f share aj (j = 1, 2, 3) CM with its k-th linear differential polynomial L[f] in D, then f = L[f]. This generalizes the corresponding results from Frank and Schwick, Zheng and Li-Liu-Yi.

Brück Conjecture with Hyper-Order $$\varvec{\sigma }_{{\mathbf {2}}}{\varvec{(}\mathbf {f}\varvec{)}} {\varvec{&lt;}} {\mathbf {1}}$$ σ 2 ( f ) &lt; 1 and Linear Differential Polynomials

In this paper, we prove a uniqueness theorem for an entire function of hyper-order $$\sigma _{2}(f)<1$$ that share z CM with its linear differential polynomial. Our result on the Bruck conjecture, which improves and generalizes some results mentioned before.

Linear differential polynomials in zero-free meromorphic functions

The paper determines all meromorphic functions f in C such that f and F have finitely many zeros, where F = f (k) + ak−1f(k−1)+ ... + a0f with k ≥ 3 and the aj rational functions. MSC 2010:30D35.

An entire function sharing fixed points with its linear differential polynomial

We study the uniqueness of entire functions, when they share a linear polynomial, in particular, fixed points, with their linear differential polynomials.

Value distribution and uniqueness results of L-functions concerning certain linear differential polynomials

In this paper, we study a uniqueness question of meromorphic functions concerning certain linear differential polynomials that share a nonzero finite value with the same of L-functions. The results in this paper extend the corresponding results from Li[6] and Li & Li[7].

English

We study the uniqueness of entire functions which share a polynomial with their linear differential polynomials.