<p>In this paper, we study the class of functions whose second difference admits a product form. In particular, we present the general solutions of the functional equation <Equation ID="Equ87"> <EquationSource Format="TEX">$\begin{aligned} \begin{aligned} f_1(x_1y_1,x_2y_2)-f_1(x_1y_1,x_2y_2^{-1})&-f_2(x_1y_1^{-1},x_2y_2)+f_2(x_1y_1^{-1},x_2y_2^{-1})\\&=g(x_1,x_2)h(y_1,y_2) \end{aligned} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <msubsup> <mi>y</mi> <mn>2</mn> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>-</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <msubsup> <mi>y</mi> <mn>1</mn> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <msubsup> <mi>y</mi> <mn>1</mn> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <msubsup> <mi>y</mi> <mn>2</mn> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f_1,f_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> are unknown functions and satisfy the conditions <Equation ID="Equ88"> <EquationSource Format="TEX">$\begin{aligned} \begin{aligned}&f_1(x_1y_1z_1,x_2)=f_1(x_1z_1y_1,x_2),f_1(x_1,x_2y_2z_2)=f_1(x_1,x_2z_2y_2),\\&f_2(x_1y_1z_1,x_2)=f_2(x_1z_1y_1,x_2),f_2(x_1,x_2y_2z_2)=f_2(x_1,x_2z_2y_2) \end{aligned} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mtable> <mtr> <mtd /> <mtd columnalign="left"> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>y</mi> <mn>1</mn> </msub> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>z</mi> <mn>1</mn> </msub> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>y</mi> <mn>2</mn> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>y</mi> <mn>1</mn> </msub> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>z</mi> <mn>1</mn> </msub> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>y</mi> <mn>2</mn> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$x_i,y_i,z_i\in G_i(i=1,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>z</mi> <mi>i</mi> </msub> <mo>∈</mo> <msub> <mi>G</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$G_i(i=1,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are arbitrary groups.</p>

A note on second differences in product form in two variables

期刊论文
发表日期：

2026年2月20日

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Ying Ying Xu,
Hou Yu Zhao

摘要

In this paper, we study the class of functions whose second difference admits a product form. In particular, we present the general solutions of the functional equation $\begin{aligned} \begin{aligned} f_1(x_1y_1,x_2y_2)-f_1(x_1y_1,x_2y_2^{-1})&-f_2(x_1y_1^{-1},x_2y_2)+f_2(x_1y_1^{-1},x_2y_2^{-1})\\&=g(x_1,x_2)h(y_1,y_2) \end{aligned} \end{aligned}$ $\begin{matrix} \begin{matrix} f_{1} (x_{1} y_{1}, x_{2} y_{2}) - f_{1} (x_{1} y_{1}, x_{2} y_{2}^{- 1}) & - f_{2} (x_{1} y_{1}^{- 1}, x_{2} y_{2}) + f_{2} (x_{1} y_{1}^{- 1}, x_{2} y_{2}^{- 1}) \\ = g (x_{1}, x_{2}) h (y_{1}, y_{2}) \end{matrix} \end{matrix}$ where $f_1,f_2$ $f_{1}, f_{2}$ are unknown functions and satisfy the conditions $\begin{aligned} \begin{aligned}&f_1(x_1y_1z_1,x_2)=f_1(x_1z_1y_1,x_2),f_1(x_1,x_2y_2z_2)=f_1(x_1,x_2z_2y_2),\\&f_2(x_1y_1z_1,x_2)=f_2(x_1z_1y_1,x_2),f_2(x_1,x_2y_2z_2)=f_2(x_1,x_2z_2y_2) \end{aligned} \end{aligned}$ $\begin{matrix} \begin{matrix} f_{1} (x_{1} y_{1} z_{1}, x_{2}) = f_{1} (x_{1} z_{1} y_{1}, x_{2}), f_{1} (x_{1}, x_{2} y_{2} z_{2}) = f_{1} (x_{1}, x_{2} z_{2} y_{2}), \\ f_{2} (x_{1} y_{1} z_{1}, x_{2}) = f_{2} (x_{1} z_{1} y_{1}, x_{2}), f_{2} (x_{1}, x_{2} y_{2} z_{2}) = f_{2} (x_{1}, x_{2} z_{2} y_{2}) \end{matrix} \end{matrix}$ for all $x_i,y_i,z_i\in G_i(i=1,2)$ $x_{i}, y_{i}, z_{i} \in G_{i} (i = 1, 2)$ , where $G_i(i=1,2)$ $G_{i} (i = 1, 2)$ are arbitrary groups.