A characterization of Cauchy kernels

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Department of Mathematical Sciences


If Φ is a function of one variable, its nth order Cauchy kernel is defined by {Mathematical expression} where ∅ ≠J {Mathematical expression}In = {1, 2,⋯, n} and xJ = ∑j ∈ Jxj. If f is a function of n variable, its ith partial Cauchy kernel of order n, {Mathematical expression}, is its Cauchy kernel of order n with respect to its ith variable with all the other variables held fixed. For n = 2 the Kurepa functional equation can be expressed by {Mathematical expression}. Here it is shown that {Mathematical expression} characterizes symmetric functions of the form f = {Mathematical expression} Φ and that the general solution of (*) is given by f = {Mathematical expression} Φ +A where A is n-multiadditive with ∑σ ∈SnA(xσ(1),⋯, xσ(n))=0.

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© 1990 Birkhäuser Verlag. Publisher’s version of record: https://doi.org/10.1007/BF02112301

Publication Title

Aequationes Mathematicae