A characterization of Cauchy kernels

Summary

If Φ is a function of one variable, itsnth order Cauchy kernel is defined by

$$\mathop K\limits_n \Phi (x_1 ,...,x_n ) = \sum\limits_{r = 1}^n {( - 1)^{n - r} \sum\limits_{\left| J \right| = r} {\Phi (x_J )} }$$

where ∅ ≠J $\subseteq$ I _n = {1, 2,⋯,n} andx _J = ∑ _{j ∈ J} x _j . Iff is a function ofn variable, itsith partial Cauchy kernel of ordern,$\mathop K\limits_n^{(i)} f$, is its Cauchy kernel of ordern with respect to its ith variable with all the other variables held fixed. Forn = 2 the Kurepa functional equation can be expressed by

$$\begin{gathered} \mathop K\limits_2^{(1)} f(x_1 ,x_2 ;x_3 ) = f(x_1 + x_2 ,x_3 ) - f(x_1 ,x_3 ) - f(x_2 ,x_3 ) \hfill \\ {\mathbf{ }} = f(x_1 ,x_2 + x_3 ) - f(x_1 ,x_2 ) - f(x_1 ,x_3 ) = \mathop K\limits_2^{(2)} f(x_1 ,x_2 ,x_3 ) \hfill \\\end{gathered} $$

. Here it is shown that

$$\mathop K\limits_2^{(i)} f = \mathop K\limits_2^{(j)} f{\mathbf{ }}for{\mathbf{ }}i,j = 1,2,...,n$$

((*))

characterizes symmetric functions of the formf =$\mathop K\limits_n$ Φ and that the general solution of (*) is given byf =$\mathop K\limits_n$ Φ +A whereA isn-multiadditive with ∑ _{σ ∈Sn} A(x _σ(1),⋯,x _σ(n) )=0.