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Periodica Mathematica Hungarica
Author:
P. Schroth
The system of functional equations
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$$\forall p\varepsilon N_ + \forall (x,y)\varepsilon D:f(x,y) = \frac{1}{p}\sum\limits_{k = 0}^{p - 1} {f(x + ky,py)}$$
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is suited to characterize the functions
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$$(x,y) \mapsto y^m B_m \left( {\frac{x}{y}} \right),m\varepsilon N,$$
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B
m
means them-th Bernoulli-polynomial,
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$$(x,y) \mapsto \exp (x)y(\exp (y) - 1)^{ - 1}$$
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(for these functionsD =R ×R
+) and
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$$(x,y) \mapsto \log y + \Psi \left( {\frac{x}{y}} \right)(D = R_ + \times R_ + )$$
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as those continuous solutions of this system which allow a certain separation of variables and take on some prescribed function values.
