Summary
Let G be an abelian group and X a vector space over the rationals. If Φ: G → X its 1st Cauchy difference is the function K2Φ:G2 → X defined by K2Φ(x 1, x2) = Φ(x1 + x2) − Φ(x1) − Φ(x2) and in general for n = 2, 3, … the (n − 1)th Cauchy difference of Φ is the function, K nΦ: Gn → X, defined by
where ϕ ≠ ⊆ I n = {1, 2, …, n} and x J = Σj∈JxJ. If Ψ: Gn → X(n = 2, 3, …) then its i-th partial Cauchy difference of order r(r = 2, 3, …), Kr (i)Ψ: Gn+r+1 → X, is its Cauchy difference of order r − 1 with respect to its i-th variable with all the other variables held fixed. For n = 2 we have K2 (1)Ψ(x1, x2, x3) = Ψ(x1 + x2, x3) − Ψ(x1, x3) − Ψ(x 2, x3) and K2 (2)Ψ(x1; x2, x3) = Ψ(x1, x2 + x3) − Ψ(x1, x2) − Ψ(x1, x3).
In this paper if ƒ = < ƒ1, … ƒn >: G n → xn the solution of K2 (i) ƒ j = K2 (j) ƒ i (i ≠ j) is given for n = 2 and 3.
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Dedicated to Professor Janos Aczél on the occasion of his 70th birthday.
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Heuvers, K.J. Cauchy-Difference Conservative Vector Fields for Dimension Two and Three. Results. Math. 26, 298–305 (1994). https://doi.org/10.1007/BF03323052
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DOI: https://doi.org/10.1007/BF03323052