This paper is devoted to investigate the existence and the forms of entire solutions of several Fermat type quadratic trinomial differential difference equations. Our results improve some results due to Liu and Yang [An. Stiint. Univ. Al. I. Cuza Iasi. Mat., 2016], Han and Lü [J. Contemp. Math. Anal., 2019], Luo, Xu and Hu [Open Math., 2021].

Entire solutions for several Fermat type differential difference equations

&lt;abstract&gt;&lt;p&gt;In this paper, we study the entire solutions of two quadratic functional equations in the complex plane. One consists of three basic terms, $ f(z), f'(z) $ and $ f(z+c) $, and the other one consists of $ f(z), f'(z) $ and $ f(qz) $. These two equations can be transformed into functional equations of Fermat-type. We prove that if these two equations admit finite order transcendental entire solutions, then the solutions of these two equations are both exponential functions, and their exponents are one degree polynomials, whose coefficients of the first degree term are closely related to the coefficients of the functional equation. Moreover, examples are given to show that the theorems are true. The feature of this paper is that the Fermat-type equations contain three quadratic terms, while the equations that have been studied in the previous articles in this field contain only two quadratic terms. The addition of $ f(qz) $ will make the proof methods in this paper very different from those in the existing literature. The proof becomes more difficult, and the number of cases that need to be discussed becomes much larger. In addition, when dealing with the analytical property of $ f $, we also use a different method from the previous literature.&lt;/p&gt;&lt;/abstract&gt; 

The exact transcendental entire solutions of complex equations with three quadratic terms

This paper concerns the existence and precise expression form of entire solutions to a certain type of delay-differential equation. The significance of our results lie in that we generalize and supplement the related results obtained recently.

/pdf/on-entire-function-solutions-to-fermat-delay-differential-2f1f2gd7.pdf

On Entire Function Solutions to Fermat Delay-Differential Equations

The main purpose of this paper is to find the explicit forms for entire solutions of two certain types of Fermat-type q-difference differential equations. Some previous results are generalized and examples are constructed to show that the results are accurate.

Entire solutions of two certain types of quadratic trinomial q-difference differential equations

On solutions of two categories of q-shift equations in two dimensional complex field

/pdf/entire-solutions-for-several-general-quadratic-trinomial-3tynl3znfg.pdf

Entire solutions for several general quadratic trinomial differential difference equations

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Entire solutions for several general quadratic trinomial differential difference equations