For meromorphic functions f and g in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb{C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, the functional equation of the form <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$af^2(z)+2\omega f(z)g(z)+bg^2(z)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <msup> <mi>f</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>ω</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>b</mi> <msup> <mi>g</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> (where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\omega^2\neq 0,ab$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ω</mi> <mn>2</mn> </msup> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$a, b\in\mathbb{C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation>) is called quadratic trinomial equation. In this paper, we investigate the existence and exact forms of solutions of quadratic trinomial partial delay differential-difference equations by utilizing Nevanlinna theory in several complex variables. Our results presented in the paper represent improvements upon some recent findings in [Rocky Mountain J. Math., 54 (2023), 1535–1550], [Mediterr. J. Math., 15 (2018), 1–14], [Anal. Math., 48 (2022), 199–226]. By exhibiting examples, we endorse the validity of conclusions of the main results. In particular, when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\omega^2=ab$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ω</mi> <mn>2</mn> </msup> <mo>=</mo> <mi>a</mi> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation>, we show that infinite order solutions of such equation also exist.

Solutions of delay differential-difference equations in $\mathbb{C}^n$

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发表日期：

2026年6月19日

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H. Y. Xu,
S. Mandal,
M. B. Ahamed

摘要

For meromorphic functions f and g in $\mathbb{C}^n$ $C^{n}$ , the functional equation of the form $af^2(z)+2\omega f(z)g(z)+bg^2(z)=1$ $a f^{2} (z) + 2 ω f (z) g (z) + b g^{2} (z) = 1$ (where $\omega^2\neq 0,ab$ $ω^{2} \neq 0, a b$ with $a, b\in\mathbb{C}$ $a, b \in C$ ) is called quadratic trinomial equation. In this paper, we investigate the existence and exact forms of solutions of quadratic trinomial partial delay differential-difference equations by utilizing Nevanlinna theory in several complex variables. Our results presented in the paper represent improvements upon some recent findings in [Rocky Mountain J. Math., 54 (2023), 1535–1550], [Mediterr. J. Math., 15 (2018), 1–14], [Anal. Math., 48 (2022), 199–226]. By exhibiting examples, we endorse the validity of conclusions of the main results. In particular, when $\omega^2=ab$ $ω^{2} = a b$ , we show that infinite order solutions of such equation also exist.

Solutions of delay differential-difference equations in \(\mathbb{C}^n\)

摘要