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Katsuya Ishizaki

Publications and source records attributed to Katsuya Ishizaki.

8 recordsLinked to original sources
arXiv2023

All possible orders less than 1 of transcendental entire solutions of linear difference equations with polynomial coefficients

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.

math.CV↗
arXiv2022

Value distribution of exponential polynomials and their role in the theories of complex differential equations and oscillation theory

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.

math.CV↗
arXiv2021

Dual exponential polynomials and a problem of Ozawa

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.

math.CV↗
arXiv2020

Binomial series and complex difference equations

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 $1/2$.

math.CV↗
arXiv2019

Exponential polynomials in the oscillation theory

Supposing that $A(z)$ is an exponential polynomial of the form $$ A(z)=H_0(z)+H_1(z)e^{ζ_1z^n}+\cdots +H_m(z)e^{ζ_mz^n}, $$ where $H_j$'s are entire and of order $<n$, it is demonstrated that the function $H_0(z)$ and the geometric location of the leading coefficients $ζ_1,\ldots,ζ_m$ play a key role in the oscillation of solutions of the differential equation $f''+A(z)f=0$. The key tools consist of value distribution properties of exponential polynomials, and elementary properties of the Phragmén-Lindelöf indicator function. In addition to results in the whole complex plane, results on sectorial oscillation are proved.

math.CV↗
arXiv2018

Mason's theorem with a difference radical

Differential calculus is not a unique way to observe polynomial equations such as $a+b=c$. We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials $a$, $b$ and $c$ satisfying the equation above. Then a difference $abc$ theorem for polynomials is proved using a new notion of a radical of a polynomial. Two results on the non-existence of polynomial solutions to difference Fermat type functional equations are given as applications. We also introduce a truncated second main theorem for differences, and use it to consider difference Fermat type equations with transcendental entire solutions.

math.CV↗
arXiv2017

Meromorphic solutions of algebraic difference equations

It is shown that the difference equation \begin{equation}\label{abseq} (Δf(z))^2=A(z)(f(z)f(z+1)-B(z)), \qquad\qquad (1) \end{equation} where $A(z)$ and $B(z)$ are meromorphic functions, possesses a continuous limit to the differential equation \begin{equation}\label{abseq2} (w')^2=A(z)(w^2-1),\qquad\qquad (2) \end{equation} which extends to solutions in certain cases. In addition, if (1) possesses two distinct transcendental meromorphic solutions, it is shown that these solutions satisfy an algebraic relation, and that their growth behaviors are almost same in the sense of Nevanlinna under some conditions. Examples are given to discuss the sharpness of the results obtained. These properties are counterparts of the corresponding results on the algebraic differential equation (2).

math.CV↗