Some new findings concerning value distribution of a pair of delay-differential polynomials

The paired Hayman's conjecture of different types are considered. More accurately speaking, the zeros of a pair of fⁿL(z,g)-a₁(z) and gᵐL(z,f)-a₂(z) are characterized using different methods from those previously employed, where f and g are both transcendental entire functions, $L(z,f)$ and $L(z,g)$ are non-zero linear delay-differential polynomials, min,m\≥ 2, a₁,a₂ are non-zero small functions with relative to f and g, or to fⁿ(z)L(z,g) and gᵐ(z)L(z,f), respectively. These results give answers to three open questions raised by Gao, Liu[Bull. Korean Math. Soc. 59 (2022)] and Liu, Liu[J. Math. Anal. Appl. 543 (2025)].