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On Cauchy differences of all orders. (English) Zbl 0744.39003
The Cauchy difference of order 1 of a mapping $$f$$ is $$f(x+y)-f(x)-f(y)$$. B. Jessen, J. Karpf and A. Thorup [Math. Scand. 22, 257–265 (1968; Zbl 0183.04004)] characterized as a Cauchy difference of order 1 the two place function $$S: G^2\to X$$ where $$G$$ is an abelian group and $$X$$ a divisible abelian group. The characterization is unique up to an additive function.
The Cauchy difference of order 2 is $$f(x+y+z)-f(x+y)-f(x+z)- f(y+z)+f(x)+f(y)+f(z)$$. The authors characterize three place functions $$\Sigma: G^3\to X$$, for $$G$$ an abelian group and $$X$$ a rational vector space, as Cauchy differences of order 2 unique up to a generalized polynomial of degree 2. They then show this result generalizes naturally but not trivially to order $$n>2$$.
In the statement of the uniqueness conditions of the generalization, the expression $$\frac{1}{n!}\sigma S_{12\dots n}$$ on line 2 of p. 145 should read $$\frac{1}{n!}\delta S_{12\ldots n}$$.

MSC:
 39B52 Functional equations for functions with more general domains and/or ranges 39A70 Difference operators
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References:
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