On entire solutions of two kinds of quadratic trinomial Fermat type functional equations

TL;DR

The paper analyzes quadratic trinomial Fermat-type equations of both differential-difference and $q$-difference types, establishing complete classifications of entire solutions under Nevanlinna theory. It shows that, for linear right-hand exponents $g(z)=\alpha z+\beta$, the solutions fall into explicit exponential, linear, or mixed exponential-polynomial families determined by the data in the matrix $\mathcal{A}$ (with $r(\mathcal{A})=2$), and, for higher-degree polynomials $g$, characterizes when solutions exist (or do not exist) with precise forms. In the differential-difference case, all admissible forms are obtained, including an exponential-modulated form $f(z) \propto e^{g-p}$ with $\deg p=\deg g-1$ when $\mathcal{A}_1=\mathcal{A}_2=0$, while in the $q$-difference setting no entire solutions exist for $\deg g>1$ under the finite $\rho_2$ growth condition. The results advance the understanding of Fermat-type functional equations by resolving Q1–Q2 in both settings and provide concrete, checkable solution templates via explicit constants and polynomial modifiers, with supporting examples.

Abstract

The existence of entire solutions to quadratic trinomial Fermat type differential-difference equations and \(q\)-difference differential equations involving second-order derivatives is studied by using Nevanlinna theory, and the exact form of entire solutions of the equations mentioned above is founded. Furthermore, some examples are given to show these results.

Theorems & Definitions (19)

• Theorem A
• Remark 1.1
• Theorem B
• Theorem C
• Example 1.1
• Theorem 2.1
• Remark 2.1
• Remark 2.2
• proof : Proof of Theorem \ref{['th1']}
• Theorem 3.1