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

TL;DR

This paper characterizes all possible orders less than 1 for transcendental entire solutions of linear difference equations with polynomial coefficients, providing conditions, classifications, and constructions for such solutions.

Contribution

It offers a complete classification of orders less than 1 for solutions, including existence conditions, possible values, and explicit constructions.

Findings

Identifies conditions for existence of solutions with order less than 1.

Lists all possible orders less than 1 for solutions.

Constructs equations with solutions of any rational order between 0 and 1.

Abstract

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)\Delta^mf(z)+\cdots+P_1(z)\Delta f(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 $0<\rho<1$, we can construct a linear difference equation with polynomial coefficients which has a transcendental entire solution of order $\rho$. At least, some examples are illustrated for our main theorems.