The functional equation <Equation ID="Equ37"> <EquationSource Format="TEX">$\begin{aligned} \psi (x) =\prod ^{n}_{j=1} \psi \left( f_j(x)\right) ^{p_j(x)} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∏</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mi>ψ</mi> <msup> <mfenced close=")" open="("> <msub> <mi>f</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mrow> <msub> <mi>p</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>occurs in connection with some characterization problems in probability theory. We find the form of its solutions defined in a vicinity of zero and fulfilling a natural asymptotic condition at zero: real-valued ones for a wide class of functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p_1, \ldots ,p_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and complex-valued solutions when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p_1= \cdots =p_n=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The obtained results generalize some theorems proved by Vincze (Magyar Tud. Akad. Mat. Kutató Int., Közl 7:357–361, 1962), Laha and Lukacs (Aequationes Math. 16:259–274, 1977), Kuczma, Choczewski and Ger (Iterative Functional Equations, Encyclopedia of Mathematics and its Applications 32, Cambridge University Press, Cambridge, 1990), and Baker (Proc. Amer. Math. Soc. 121:767–773, 1994). As a consequence we obtain an extension of a result by Zdun (Aequationes Math. 8:229–232, 1972). It provides a new characterization of the complex exponential functions. We record also the form of complex-valued solutions of the equation <Equation ID="Equ38"> <EquationSource Format="TEX">$\begin{aligned} \varphi (x) =\sum ^{n}_{j=1} p_j(x)\varphi \left( f_j(x)\right) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>p</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>φ</mi> <mfenced close=")" open="("> <msub> <mi>f</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with some asymptotics at zero.

On two functional equations of finite order and their complex-valued solutions with prescribed asymptotics

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2025年8月26日

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Justyna Jarczyk,
Witold Jarczyk

摘要

The functional equation $\begin{aligned} \psi (x) =\prod ^{n}_{j=1} \psi \left( f_j(x)\right) ^{p_j(x)} \end{aligned}$ $\begin{matrix} ψ (x) = \prod_{j = 1}^{n} ψ {(f_{j}, (x))}^{p_{j} (x)} \end{matrix}$ occurs in connection with some characterization problems in probability theory. We find the form of its solutions defined in a vicinity of zero and fulfilling a natural asymptotic condition at zero: real-valued ones for a wide class of functions $p_1, \ldots ,p_n$ $p_{1}, \dots, p_{n}$ and complex-valued solutions when $p_1= \cdots =p_n=1$ $p_{1} = \dots = p_{n} = 1$ . The obtained results generalize some theorems proved by Vincze (Magyar Tud. Akad. Mat. Kutató Int., Közl 7:357–361, 1962), Laha and Lukacs (Aequationes Math. 16:259–274, 1977), Kuczma, Choczewski and Ger (Iterative Functional Equations, Encyclopedia of Mathematics and its Applications 32, Cambridge University Press, Cambridge, 1990), and Baker (Proc. Amer. Math. Soc. 121:767–773, 1994). As a consequence we obtain an extension of a result by Zdun (Aequationes Math. 8:229–232, 1972). It provides a new characterization of the complex exponential functions. We record also the form of complex-valued solutions of the equation $\begin{aligned} \varphi (x) =\sum ^{n}_{j=1} p_j(x)\varphi \left( f_j(x)\right) \end{aligned}$ $\begin{matrix} φ (x) = \sum_{j = 1}^{n} p_{j} (x) φ (f_{j}, (x)) \end{matrix}$ with some asymptotics at zero.