Transcendental entire solutions of some delay differential equations

The primary focus of this study was to analyze the following three delay differential equations:

$$\omega (z+1)-\omega (z-1)+a(z)\frac{{\omega }^{(k)}(z)}{\omega (z)}=R(z,\omega (z))=\frac{P(z,\omega )}{Q(z,\omega )},$$

$$\omega (z+1)-\omega (z-1)+a(z)(\frac{{\omega }^{\prime }(z)}{\omega (z)}{)}^k=R(z,\omega (z))=\frac{P(z,\omega )}{Q(z,\omega )}$$

and

$$\omega (z+1)+a(z)\frac{{\omega }^{(k)}(z)}{\omega (z)}=R(z,\omega (z))=\frac{P(z,\omega )}{Q(z,\omega )},$$

where $P(z,\omega )$ and $Q(z,\omega )$ were polynomials in $\omega$ with rational coefficients in $z$ and no common roots, $a(z)$ was a rational function, and $k\ge 1$ was an integer. This paper investigated the forms of the transcendental entire solutions for the above equations. Some examples were given to support these results.