The growth of transcendental entire solutions of linear difference equations with polynomial coefficients

TL;DR

This work addresses the growth of transcendental entire solutions to linear difference equations with polynomial coefficients. It leverages binomial-series representations and a Newton-polygon–style balance to obtain precise growth laws for solutions of order $<1$, including explicit type constants. It proves a constructive result: for any rational order in $(0,1)$ and any positive mean type, there exists a difference equation with polynomial coefficients possessing a transcendental entire solution of that order and type. The authors illustrate the theory with explicit examples (e.g., orders $1/2$, $1/3$, and $3/4$) and show how to realize prescribed growth characteristics, improving prior results in the literature.

Abstract

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.

Figures (1)

• Figure 1:

Theorems & Definitions (10)

• Theorem 2.1
• Corollary 2.2
• Theorem 2.3
• Theorem 3.1
• Example 3.2
• Example 3.3
• Example 3.4
• Example 6.1
• Example 6.2
• Example 6.3