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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 \(\frac{1}{n!}\delta S_{12\ldots n}\).

39B52 Functional equations for functions with more general domains and/or ranges
39A70 Difference operators
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[1] Djoković, D. Z.,A representation theorem for (X 1 1)X n 1) and its applications. In Ann. Polonici Math.22 (1969), 189–198. · Zbl 0187.39903
[2] Ebanks, B. R.,Kurepa’s functional equation on semigroups. Stochastica6 (1982), 39–55. · Zbl 0524.39010
[3] Ebanks, B. R.,Kurepa’s functional equation on Gaussian semigroups. InFunctional Equations: History, Applications and Theory (ed. J. Aczél). Reidel (Kluwer), Dordrecht–Boston–Lancaster, 1984, pp. 167–173.
[4] Erdös, J.,A remark on the paper ”On some functional equations” by S. Kurepa. Glasnik Mat.-Fiz. i Astron.14 (1959), 3–5.
[5] Gajda, Z.,A solution to a problem of J. Schwaiger. Aequationes Math.32 (1987), 38–44. · Zbl 0619.39006
[6] Heuvers, K. J.,A characterization of Cauchy kernels. Aequationes Math.40 (1990), 281–306. · Zbl 0716.39004
[7] Hosszú, M.,On the functional equation F(x + y, z) + F(x, y)=F(x, y + z) + F(y, z). Period. Math. Hungar.1 (1971), 213–216. · Zbl 0235.39005
[8] Jessen, B., Karpf, J. andThorup, A.,Some function equations in groups and rings. Math. Scand.22 (1968), 257–265. · Zbl 0183.04004
[9] Ng, C. T.,Representation for measures of information with the branching property. Inform. and Control25 (1974), 45–56. · Zbl 0279.94018
[10] Ng, C. T.,Remark 6 to Problem 5 (ii) of I. Fenyö. Aequationes Math.26 (1984), 262–263.
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