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Borel exceptional values in several complex variables and their applications to shared values of shifts and difference operators

Abhijit Banerjee, Sujoy Majumder, Jhilik Banerjee

Abstract

In this paper, we investigate shared value problems for shifts and higher-order difference operators of meromorphic and entire functions in several complex variables. Using Nevanlinna theory in $\mathbb{C}^n$, we obtain new uniqueness theorems when functions share values counting or ignoring multiplicities, extending several classical one-variable results to higher dimensions. A key contribution of this work appears in Section 2, where we establish fundamental results on Borel exceptional values in several complex variables. These propositions provide the main tools for proving our principal theorems. As applications, we derive conditions ensuring that a transcendental entire function satisfies $Δ_c^{k} \equiv d\hspace{.05cc} f$ and we study meromorphic solutions of certain partial differential-difference equations, obtaining growth estimates and structural descriptions of entire solutions. To the best of our knowledge, this is the first systematic study of such shared value problems for higher-order difference operators in several complex variables.

Borel exceptional values in several complex variables and their applications to shared values of shifts and difference operators

Abstract

In this paper, we investigate shared value problems for shifts and higher-order difference operators of meromorphic and entire functions in several complex variables. Using Nevanlinna theory in , we obtain new uniqueness theorems when functions share values counting or ignoring multiplicities, extending several classical one-variable results to higher dimensions. A key contribution of this work appears in Section 2, where we establish fundamental results on Borel exceptional values in several complex variables. These propositions provide the main tools for proving our principal theorems. As applications, we derive conditions ensuring that a transcendental entire function satisfies and we study meromorphic solutions of certain partial differential-difference equations, obtaining growth estimates and structural descriptions of entire solutions. To the best of our knowledge, this is the first systematic study of such shared value problems for higher-order difference operators in several complex variables.
Paper Structure (8 sections, 17 theorems, 122 equations, 1 table)

This paper contains 8 sections, 17 theorems, 122 equations, 1 table.

Table of Contents

  1. Preliminaries on Nevanlinna Theory in Several Complex Variables
  2. Borel exceptional value and related results in several complex variables
  3. Shared Value Problems Associated with Shifts
  4. Auxiliary lemmas
  5. Proof of Theorem \ref{['t2.2']}
  6. Proof of Theorem \ref{['t2.3']}
  7. Proof of Theorem \ref{['t2.4']}
  8. Proof of Theorem \ref{['t2.5']}

Key Result

Proposition 2.1

Let $f$ be a transcendental meromorphic function in $\mathbb{C}^n$ with finite positive order $\rho(f)$. Then $f$ has at most two Borel exceptional values.

Theorems & Definitions (35)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 25 more