#### Title

A characterization of Cauchy kernels

#### Document Type

Article

#### Publication Date

12-1990

#### Department

Department of Mathematical Sciences

#### Abstract

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.

#### Publication Title

Aequationes Mathematicae

#### Recommended Citation

Heuvers, K.
(1990).
A characterization of Cauchy kernels.
*
Aequationes Mathematicae,
40*(1), 281-306.
http://doi.org/10.1007/BF02112301

Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/4339

## Publisher's Statement

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