We investigate the functional equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f(f(x+y))=f(x+y)+f(x)f(y),\ x,y\in \mathbb R.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</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> <mo>,</mo> <mspace width="4pt" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Given a real number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$c\ne 0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> it was shown in [<CitationRef CitationID="CR5">5</CitationRef>] that there is a non-trivial solution f with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f(0)=c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation> iff c is transcendental and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$>-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>></mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In that case all values of f are transcendental. In this paper we find all solutions of the functional equation, discuss their odd properties and the connection with ideals in a group algebra.

On the solutions of a functional equation attaining transcendental numbers only

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2026年5月14日

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Johannes Schoißengeier

摘要

We investigate the functional equation $f(f(x+y))=f(x+y)+f(x)f(y),\ x,y\in \mathbb R.$ $f (f (x + y)) = f (x + y) + f (x) f (y), x, y \in R .$ Given a real number $c\ne 0,$ $c \neq 0,$ it was shown in [5] that there is a non-trivial solution f with $f(0)=c$ $f (0) = c$ iff c is transcendental and $>-1$ $> - 1$ . In that case all values of f are transcendental. In this paper we find all solutions of the functional equation, discuss their odd properties and the connection with ideals in a group algebra.