arXiv subjects
Publications and source records attributed to Zhi-Tao Wen.
In this paper, we give a new definition of the $q$-weight of zeros, which reduces to the multiplicity of zeros as $q\to 1$. Furthermore, we obtain a $q$-difference version of the Stothers-Mason theorem by means of the new definition of the $q$-difference radical, which covers the classical Stothers-Mason theorem as $q\to 1$. As applications, we study the polynomial solutions of $q$-difference Fermat type functional equations.
In this paper, we study the growth of transcendental entire solutions of linear difference equations \begin{equation} P_m(z)Δ^mf(z)+\cdots+P_1(z)Δf(z)+P_0(z)f(z)=0,\tag{+} \end{equation} where $P_j(z)$ are polynomials for $j=0,\ldots,m$. At first, we reveal type of binomial series in terms of its coefficients. Second, we give a list of all possible orders, which are less than 1, and types of transcendental entire solutions of linear difference equations $(+)$. In particular, we give so far the best precise growth estimate of transcendental entire solutions of order less than 1 of $(+)$, which improves results in [3, 4], [5], [7]. Third, for any given rational number $ρ\in(0,1)$ and real number $σ\in(0,\infty)$, we can construct a linear difference equation with polynomial coefficients which has a transcendental entire solution of order $ρ$ and type $σ$. At last, some examples are illustrated for our main theorem.
In this paper, we focus on the difference analogue of the Stothers-Mason theorem for entire functions of order less than 1, which can be seen as difference $abc$ theorem for entire functions. We also obtain the difference analogue of truncated version of Nevanlinna second main theorem which reveals that a subnormal meromorphic function $f(z)$ such that $Δf(z)\not\equiv 0$ cannot have too many points with long length in the complex plane. Both theorems depend on new definitions of the length of poles and zeros of a given meromorphic function in a domain. As for the application, we consider entire solutions of difference Fermat functional equations.
We show a necessary and sufficient condition on the existence of finite order entire solutions of linear differential equations $$ f^{(n)}+a_{n-1}f^{(n-1)}+\cdots+a_1f'+a_0f=0,\eqno(+) $$ where $a_i$ are exponential sums for $i=0,\ldots,n-1$ with all positive (or all negative) rational frequencies and constant coefficients. Moreover, under the condition that there exists a finite order solution of (+) with exponential sum coefficients having rational frequencies and constant coefficients, we give the precise form of all finite order solutions, which are exponential sums. It is a partial answer to Gol'dberg-Ostrovskiǐ Problem and Problem 5 in \cite{HITW2022} since exponential sums are of completely regular growth.
In this paper, we study all possible orders which are less than 1 of transcendental entire solutions of linear difference equations \begin{equation} P_m(z)Δ^mf(z)+\cdots+P_1(z)Δf(z)+P_0(z)f(z)=0,\tag{+} \end{equation} where $P_j(z)$ are polynomials for $j=0,\ldots,m$. Firstly, we give the condition on existence of transcendental entire solutions of order less than 1 of difference equations (+). Secondly, we give a list of all possible orders which are less than 1 of transcendental entire solutions of difference equations (+). Moreover, the maximum number of distinct orders which are less than 1 of transcendental entire solutions of difference equations (+) are shown. In addition, for any given rational number $0<ρ<1$, we can construct a linear difference equation with polynomial coefficients which has a transcendental entire solution of order $ρ$. At least, some examples are illustrated for our main theorems.
An exponential polynomial is a finite linear sum of terms $P(z)e^{Q(z)}$, where $P(z)$ and $Q(z)$ are polynomials. The early results on the value distribution of exponential polynomials can be traced back to Georg Pólya's paper published in 1920, while the latest results have come out in 2021. Despite of over a century of research work, many intriguing problems on value distribution of exponential polynomials still remain unsolved. The role of exponential polynomials and their quotients in the theories of linear/non-linear differential equations, oscillation theory and differential-difference equations will also be discussed. Thirteen open problems are given to motivate the readers for further research in these topics.
Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an exponential polynomial solution $f$, then the order of $f$ and of the dominant coefficient are equal, and the two functions possess a certain duality property. The results presented in this paper improve earlier results by some of the present authors, and the paper adjoins with two open problems.
The growth of meromorphic solutions of linear difference equations containing Askey-Wilson divided difference operators is estimated. The $φ$-order is used as a general growth indicator, which covers the growth spectrum between the logarithmic order $ρ_{\log}(f)$ and the classical order $ρ(f)$ of a meromorphic function $f$.
In this paper, we study the shifting zeros with its heights and an analogue to difference radical. We focus on the Stothers-Mason theorem by using falling factorials. As applications, we discuss the difference version of the Fermat type functional equations. Some examples are given.
The $φ$-order was introduced in 2009 for meromorphic functions in the unit disc, and was used as a growth indicator for solutions of linear differential equations. In this paper, the properties of meromorphic functions in the complex plane are investigated in terms of the $φ$-order, which measures the growth of functions between the classical order and the logarithmic order. Several results on value distribution of meromorphic functions are discussed by using the $φ$-order and the $φ$-exponent of convergence. Instead of linear differential equations, the applications in the complex plane lie in linear $q$-difference equations.
The Airy integral is a well-known contour integral solution of Airy's equation which has several applications and which has been used for mathematical illustrations due to its interesting properties. We present and derive properties of two families of contour integral solutions of linear differential equations, where one family includes the Airy integral and Airy's equation, such that the family generalizes known properties of the Airy integral which include exponential decay growth in a certain sector. The second family includes a known example and contains a subfamily with interesting properties where a separate analysis of three pairwise linearly independent contour integral solutions of a particular equation is given.
We consider properties of binomial series $\sum_{n=0}^\infty a_n z^{\underline{n}}$, where $z^{\underline{n}}=z(z-1)\cdots(z-n+1)$ and the convergence of binomial series in the complex domain. The order of growth of entire and meromorphic solutions of some difference equations represented by binomial series are discussed. Examples are given. As an application, we construct a difference Riccati equation possessing a transcendental meromorphic solution of order $1/2$.
The well-known $\mathcal{E}$-sets introduced by Hayman in 1960 are collections of Euclidean discs in the complex plane with the following property: The set of angles $θ$ for which the ray $\arg(z)=θ$ meets infinitely many discs in a given $\mathcal{E}$-set has linear measure zero. An important special case of an $\mathcal{E}$-set is known as the $R$-set. These sets appear in numerous papers in the theories of complex differential and functional equations. This paper offers a continuous transition from $\mathcal{E}$-sets to $R$-sets, and then to much thinner sets. In addition to rays, plane curves that originate from the zero distribution theory of exponential polynomials will be considered. It turns out that almost every such curve meets at most finitely many discs in the collection in question. Analogous discussions are provided in the case of the unit disc $\mathbb{D}$, where the curves tend to the boundary $\partial\mathbb{D}$ tangentially or non-tangentially. Finally, these findings will be used for improving well-known estimates for logarithmic derivatives, logarithmic differences and logarithmic $q$-differences of meromorphic functions, as well as for improving standard results on exceptional sets.
Differential equations of the form $f'' + A(z)f' + B(z)f = 0$ (*) are considered, where $A(z)$ and $B(z) \not\equiv 0$ are entire functions. The Lindelöf function is used to show that for any $ρ\in (1/2, \infty)$, there exists an equation of the form (*) which possesses a solution $f$ of order $ρ$ with a Nevanlinna deficient value at $0$, where $f, A(z), B(z)$ satisfy a common growth condition. It is known that such an example cannot exist when $ρ\leq 1/2$. For smaller growth functions, a geometrical modification of an example of Anderson and Clunie is used to show that for any $ρ\in (2, \infty)$, there exists an equation of the form (*) which possesses a solution $f$ of logarithmic order $ρ$ with a Valiron deficient value of at $0$, where $f, A(z), B(z)$ satisfy an analogous growth condition. This result is essentially sharp. In both proofs, the separation of the zeros of the indicated solution plays a key role. Observations on the deficient values of solutions of linear differential equations are also given, which include a discussion of Wittich's theorem on Nevanlinna deficient values, a modified Wittich theorem for Valiron deficient values, consequences of Gol'dberg's theorem, and examples to illustrate possibilities that can occur.
An exponential polynomial of order $q$ is an entire function of the form $$ f(z)=P_1(z)e^{Q_1(z)}+\cdots +P_k(z)e^{Q_k(z)}, $$ where the coefficients $P_j(z),Q_j(z)$ are polynomials in $z$ such that $$ \max\{°{Q_j}\}=q. $$ In 1977 Steinmetz proved that the zeros of $f$ lying outside of finitely many logarithmic strips around so called critical rays have exponent of convergence $\leq q-1$. This result does not say nothing about the zero distribution of $f$ in each individual logarithmic strip. Here, it is shown that the asymptotic growth of the non-integrated counting function of zeros of $f$ is asymptotically comparable to $r^q$ in each logarithmic strip. The result generalizes the first order results by Pólya and Schwengeler from the 1920's, and it shows, among other things, that the critical rays of $f$ are precisely the Borel directions of order $q$ of $f$. The error terms in the asymptotic equations for $T(r,f)$ and $N(r,1/f)$ originally due to Steinmetz are also improved.