What question did this study set out to answer?

The research aims to establish the uniqueness of finite-order meromorphic solutions for specific Painlevé equations.

March 10, 2026

On Uniqueness of Meromorphic Solutions to Some Differential-Difference Equations

Key Points

The research aims to establish the uniqueness of finite-order meromorphic solutions for specific Painlevé equations.
Analyzed differential-difference Painlevé III and V equations
Proved uniqueness under shared complex multiplicities
Considered small function behaviors of involved terms
Proved that if one meromorphic function shares certain values with another, they must be identical
Demonstrated a sufficient condition for Painlevé V involving shifted arguments
Identified conditions under which solutions are determined uniquely by their behavior at specific points.

Abstract

In this paper, we prove the uniqueness of finite-order transcendental meromorphic solutions of differential-difference Painlevé III and V equations: (z+1) (z-1) +a (z) ^ (z) (z) =₌=₀^₃a₌^{m (z) }₍=₀^{2b₍^n (z) }, and ( (z) (z+1) -1) ( (z) (z-1) -1) +a (z) ^ (z) (z) =₌=₀^₆a₌^{m (z) } (z) -b₁ (z), where a₌, b₍ and a (z) are small functions of solution f (z). We show that if the solution f (z) shares e₁, e₂, and CM with another meromorphic function g (z), then f (z) g (z). Moreover, for Painlevé V equation, if g (z) is replaced by f (z+c), it is sufficient for f (z) and f (z+c) to share the values e₁ and e₂ CM. MSC2020 numbers: 30D35.

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