caa a22 4500 465825311 CHVBK 20180323112147.0 cr unu---uuuuu 170327e19901201xx s 000 0 eng 10.1007/BF02112301 doi (NATIONALLICENCE)springer-10.1007/BF02112301 Heuvers K. Department of Mathematics, Michigan Technological University, 49931, Houghton, MI, USA aut A characterization of Cauchy kernels [Elektronische Daten] [K. Heuvers] 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. Birkhäuser Verlag, 1990 Primary 39B40, 39A70 nationallicence aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 281-306 0001-9054 40:1<281 1990 40 10 https://doi.org/10.1007/BF02112301 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02112301 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Heuvers K. Department of Mathematics, Michigan Technological University, 49931, Houghton, MI, USA aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 281-306 0001-9054 40:1<281 1990 40 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer