Differential-difference Polynomials Research Articles

Results on uniqueness of entire functions related to differential-difference polynomial

In the paper, we use the idea of normal family to investigate the uniqueness problems of entire functions when certain types of differential-difference polynomials generated by them sharing a non-zero polynomial. Also we exhibit one example to show that the conditions of our results are the best possible.

Uniqueness and variable sharing of q-shift difference polynomials of entire functions

In this paper, we study weighted sharing and characteristics of q-shift difference polynomials of entire functions and obtain a uniqueness theorem concerning weighted z-sharing of certain q-shift difference-differential polynomials of entire functions.

Uniqueness of some differential-difference polynomials of entire functions sharing small function

In this article, we discuss the uniqueness problems of differential-difference polynomials of hyper order $(\geq 1)$ meromorphic functions sharing a small function and prove some results, which generalise and improve the results due to Y. Liu, J. P. Wang and F. H. Liu.

Uniqueness of Difference-Differential Polynomials of Meromorphic Functions

We study the problems of uniqueness of difference-differential polynomials of finite-order meromorphic functions sharing a small function (ignoring multiplicities) and obtain some results that extend the results of K. Liu, X. L. Liu, and T. B. Cao.

Bivariate Kolchin-type dimension polynomials of non-reflexive prime difference-differential ideals. The case of one translation

We use the method of characteristic sets with respect to two term orderings to prove the existence and obtain a method of computation of a bivariate Kolchin-type dimension polynomial associated with a non-reflexive difference-differential ideal in the algebra of difference-differential polynomials with several basic derivations and one translation. In particular, we obtain a new proof and a method of computation of the dimension polynomial of a non-reflexive prime difference ideal in the algebra of difference polynomials over an ordinary difference field. As a consequence, it is shown that the reflexive closure of a prime difference polynomial ideal is the inverse image of this ideal under a power of the basic translation. We also discuss applications of our results to the analysis of systems of algebraic difference-differential equations.

Zero Distribution on Differential-Difference Polynomials of Exponential Type Functions

Exponential type functions are important subclasses of transcendental entire functions. In this paper, we will use some results given by Steinmetz (Manuscr Math 26:155–167, 1978) to consider the zeros of difference or differential-difference polynomials of exponential polynomials. In addition, we also consider the zeros of difference polynomials of exponential type functions with infinite order.

Value distribution of higher order differential-difference polynomial of an entire function

Value distribution of higher order differential-difference polynomial of an entire function

Value distribution of general differential-difference polynomials of meromorphic functions

In this paper, by reducing the conditions on the zeros and poles of a finite order meromorphic function f(z), we improve the results by Zheng and Chen (Acta. Math. Sin. 54, 983–992 [12]) and Zheng and Xu (Abstr. Appl. Anal. 12 pages [13]).

Some Results on the Deficiencies of Some Differential-Difference Polynomials of Meromorphic Function

For a transcendental meromorphic function f ( z ) , the main aim of this paper is to investigate the properties on the zeros and deficiencies of some differential-difference polynomials. Some results about the deficiencies of some differential-difference polynomials concerning Nevanlinna deficiency and Valiron deficiency are obtained, which are a generalization of and improvement on previous theorems given by Liu, Lan and Zheng, etc.

Meromorphic solutions of certain type of differential-difference equations

We consider meromorphic solutions to general differential-difference equation of type L ( z , f ) + ∑ j = 1 q c j ( z ) f n j ( z + ζ j ) = ∑ j = 1 r γ j e λ j z , where L ( z , f ) ≢ 0 is a linear differential-difference polynomial of f with small function coefficients, and 1 < n 1 < ⋯ < n q , γ j ( 1 ≤ j ≤ r ) are nonvanishing constants, c j ( z ) ( 1 ≤ j ≤ q ) are nonvanishing small functions. If the equation admits a transcendental meromorphic solution f of finite order satisfying that λ ( f ) < σ ( f ) , λ ( 1 / f ) < σ ( f ) , then representation of f and relations between coefficients in ψ ( z , f ) and γ i , λ j can be obtained.

Value Distribution of Differential-Difference Polynomials of Meromorphic Functions

We study the deficiencies of differential-difference polynomials. These results can be regarded as differential-difference analogs of some classical theorems on differential polynomials. In particular, an exact estimate of the deficiency is presented for some differential-difference polynomials. We also give examples showing that these results are best possible in a certain sense.

The zeros on complex differential-difference polynomials of certain types

In this paper, we consider the zeros distribution of f(z)P(z,f) -q(z), where P(z,f) is a linear differential-difference polynomial of a finite-order transcendental entire function f(z), and q(z) is a nonzero polynomial. To a certain extent, Theorem 1.1 generalizes the recent results (Latreuch and Belaïdi in Arab. J. Math. 7(1):27–37, 2018; Lü et al. in Kodai Math. J. 39(3):500–509, 2016) related to Hayman conjecture (Hayamn in Ann. Math. 70:9–42, 1959).

Uniqueness of certain type of differential-difference and difference polyniminals

In this paper we consider certain difference and differential-difference polynomials sharing some polynomial and improve a number of results in \cite{8a}, \cite{8aa} and \cite{18}. In particular we point out a gap in the argument in the proof of the main results in \cite {8aa} and rectifying the same we improve and extend the result to a large extent.

On the uniqueness of certain types of differential-difference polynomials

The purpose of the paper is to study the uniqueness problems of certain types of differential-difference polynomials sharing a small function or polynomial of certain degree under relaxed sharing hypothesis. We not only point out some gaps in the proof of the main results in [18] and [13], but also rectifying the errors, present our main results in a more compact way. In a nutshell, our results improve and generalize and rectify a recent result due to Zhang and Yi [18] and Liu and Yi [13]. Finally we indicate some questions for future research.

Some results on differential-difference analogues of Brück conjecture

Abstract In this paper, we will consider value sharing problems on a transcendental entire function f with its differential-difference polynomials. We establish some results that can be viewed as the differential-difference analogues of Brück conjecture.

The Properties of Differential-Difference Polynomials

The main aim of the paper is to improve some classical results on the distribution of zeros for differential polynomials and differential-difference polynomials. We present some results on the distribution of zeros of [f(z) n f(z + c)](k) − α(z) and [f(z) n (f(z + c) − f(z))](k) − α(z) and give some examples to show that the results are best possible in a certain sense.

On the uniqueness of certain type of differential-difference polynomials sharing a polynomial

AbstractThe purpose of the paper is to study uniqueness problems of certain types of differential-difference polynomials sharing a nonzero polynomial of certain degree under relaxed sharing hypotheses. We not only point out some gaps in the proof of the main results in [

Uniqueness of difference-differential polynomials of entire functions sharing one value

In this paper, we study the uniqueness of difference-differential polynomials of entire functions $f$ and $g$ sharing one value with counting multiplicity. In this paper we extend and generalize the results of X. Y. Zhang, J. F. Chen and W. C. Lin [17] L. Kai, L. Xin-ling and C. Ting-bin [7] and many others [2, 16].

On differential-difference polynomials

On differential-difference polynomials

Results on the deficiencies of some differential-difference polynomials of meromorphic functions

Abstract In this paper, we study the relation between the deficiencies concerning a meromorphic function f(z), its derivative f′(z) and differential-difference monomials f(z) m f(z+c)f′(z), f(z+c) n f′(z), f(z) m f(z+c). The main results of this paper are listed as follows: Let f(z) be a meromorphic function of finite order satisfying lim sup r → + ∞ T ( r , f ) T ( r , f ′ ) &lt; + ∞ , $$\mathop {\lim \,{\rm sup}}\limits_{r \to + \infty } {{T(r,\,f)} \over {T(r,\,f')}}{\rm{ &amp;#x003C; }} + \infty ,$$ and c be a non-zero complex constant, then δ(∞, f(z) m f(z+c)f′(z))≥δ(∞, f′) and δ(∞,f(z+c) n f′(z))≥ δ(∞, f′). We also investigate the value distribution of some differential-difference polynomials taking small function a(z) with respect to f(z).