Transcendental entire solutions of some delay differential equations

Changwen Peng; Huawei Huang; Mengting Xia; Changwen Peng; Huawei Huang; Mengting Xia

doi:10.3934/math.2026929

AIMS Mathematics

2026, Volume 11, Issue 7: 23050-23065. doi: 10.3934/math.2026929

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Research article

Transcendental entire solutions of some delay differential equations

1.
School of Science, Guiyang University, Guiyang 550005, China
2.
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China
3.
Guizhou Key Laboratory of NewGen Cyberspace Security, Guiyang 550001, China
4.
Guizhou Provincial Specialized Key Laboratory of Information Security Technology in Higher Education Institutions, Guiyang 550025, China

Received: 03 May 2026 Revised: 28 June 2026 Accepted: 02 July 2026 Published: 29 July 2026
MSC : 30D35, 34K40, 34M55

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'(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.
- delay differential equation,
- entire solutions,
- Painlevé equations,
- Nevanlinna theory
Citation: Changwen Peng, Huawei Huang, Mengting Xia. Transcendental entire solutions of some delay differential equations[J]. AIMS Mathematics, 2026, 11(7): 23050-23065. doi: 10.3934/math.2026929

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Abstract

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'(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.

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