Some new findings concerning value distribution of a pair of delay-differential polynomials
Jianren Long, Xuxu Xiang
TL;DR
This work investigates a paired Hayman-type value distribution for a pair of delay-differential polynomials associated with two transcendental entire functions $f$ and $g$. Using Nevanlinna theory, Weierstrass factorization, and growth relations, the authors prove that under natural conditions with $\min\{n,m\}\ge 2$, the two expressions $f^nL(z,g)$ and $g^mL(z,f)$ cannot share a nonzero generalized Picard exceptional small function $\alpha$, thereby addressing open questions of Gao–Liu and Liu–Liu. When such a shared exceptional small function exists, the paper shows it forces a rigid exponential structure on $f$ and $g$, namely $f(z)=A(z)e^{B(z)}$ and $g(z)=A_4(z)e^{-nB(z)}+A_3(z)e^{-B(z)}$, with $A,A_3,A_4$ small, and ties the exceptional values to the products $A^nA_4$ and $AA_3$. The results extend previous delay-differential analyses to paired polynomials and refine earlier claims by incorporating growth conditions, thereby contributing to the broader understanding of value distribution in delay-differential settings.
Abstract
The paired Hayman's conjecture of different types are considered. More accurately speaking, the zeros of a pair of $f^nL(z,g)-a_1(z)$ and $g^mL(z,f)-a_2(z)$ 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, $\min\{n,m\}\ge 2$, $a_1,a_2$ are non-zero small functions with relative to $f$ and $g$, or to $f^n(z)L(z,g)$ and $g^m(z)L(z,f)$, 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)].