We introduce the functional equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$g(xyz) - g(x)g(yz) - g(y)g(xz) - g(z)g(xy) + 2g(x)g(y)g(z) = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for an unknown function g mapping a semigroup S into a field K. It seems reasonable to call this a cosine functional equation because when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$S = ({\mathbb R},+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K = {\mathbb R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> the function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$g = \cos $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>=</mo> <mo>cos</mo> </mrow> </math></EquationSource> </InlineEquation> is a solution. It is not very surprising to find that this equation has a strong connection with the sine addition formula. We show that for any solution g there exists a function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f:S \rightarrow K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f(xy) = f(x)g(y) + g(x)f(y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$x,y \in S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation>. The converse is true if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$f \ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. For the case <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$K = {\mathbb C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> we show that all solutions of the cosine equation are arithmetic means of two multiplicative functions. Some more general equations are also solved.

A new cosine functional equation

期刊论文
发表日期：

2025年9月23日

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Bruce Ebanks

摘要

We introduce the functional equation $g(xyz) - g(x)g(yz) - g(y)g(xz) - g(z)g(xy) + 2g(x)g(y)g(z) = 0$ $g (x y z) - g (x) g (y z) - g (y) g (x z) - g (z) g (x y) + 2 g (x) g (y) g (z) = 0$ for an unknown function g mapping a semigroup S into a field K. It seems reasonable to call this a cosine functional equation because when $S = ({\mathbb R},+)$ $S = (R, +)$ and $K = {\mathbb R}$ $K = R$ the function $g = \cos $ $g = cos$ is a solution. It is not very surprising to find that this equation has a strong connection with the sine addition formula. We show that for any solution g there exists a function $f:S \rightarrow K$ $f : S \to K$ such that $f(xy) = f(x)g(y) + g(x)f(y)$ $f (x y) = f (x) g (y) + g (x) f (y)$ for all $x,y \in S$ $x, y \in S$ . The converse is true if $f \ne 0$ $f \neq 0$ . For the case $K = {\mathbb C}$ $K = C$ we show that all solutions of the cosine equation are arithmetic means of two multiplicative functions. Some more general equations are also solved.