Meromorphic solutions of a first order differential equations with delays

Chen, Yu; Cao, Tingbin

Comptes Rendus. Mathématique, Tome 360 (2022) no. G6, pp. 665-678.

The main purpose of this paper is to study meromorphic solutions of the first order differential equations with delays

$$w(z+1)-w(z-1)+a\left(z\right){\left(\frac{w^{\prime }\left(z\right)}{w\left(z\right)}\right)}^k=R(z,w\left(z\right))$$

and

$$w(z+1)+a\left(z\right){\left(\frac{w^{\prime }\left(z\right)}{w\left(z\right)}\right)}^k=R(z,w\left(z\right)),$$

where $k$ is a positive integer, $a\left(z\right)$ is a rational function, $R(z,w)$ is rational in $w$ with rational coefficients. Some necessary conditions on the degree of $R(z,w)$ are obtained for the equation to admit a transcendental meromorphic solution of minimal hypertype. These are extensions of some previous results due to Halburd, Korhonen, Liu and others. Some examples are given to support our conclusions.

Classification : 34K40, 30D35, 34M55

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