caa a22 4500 605508097 CHVBK 20210128100635.0 cr unu---uuuuu 210128e20150201xx s 000 0 eng 10.1007/s00010-014-0325-z doi (NATIONALLICENCE)springer-10.1007/s00010-014-0325-z Ng Che Department of Pure Mathematics, University of Waterloo, N2L 3G1, Waterloo, ON, Canada aut Kernels of higher order Cauchy differences on free groups [Elektronische Daten] [Che Ng] Let (G, .) be a group, (H, +) an abelian group and f : G → H. The first and the second Cauchy differences of f are $$\begin{aligned} & \quad C^{1}f(x,y) = f(xy) - f(x) - f(y), \\ & C^{2}f(x,y,z) = f(xyz) - f(xy) - f(yz) - f(xz) + f(x) + f(y) + f(z).\end{aligned}$$ C 1 f ( x , y ) = f ( x y ) - f ( x ) - f ( y ) , C 2 f ( x , y , z ) = f ( x y z ) - f ( x y ) - f ( y z ) - f ( x z ) + f ( x ) + f ( y ) + f ( z ) . Higher order Cauchy differences $${C^{m}f}$$ C m f are defined recursively. The functional equation $$C^{m}f = 0$$ C m f = 0 is studied. Some earlier results on the equation $${C^{2}f = 0}$$ C 2 f = 0 are extended to higher m. For m = 3 we present its general solution on free groups G. When the free group has just one generator the solution is obtained for all m. Springer Basel, 2014 Functional equations nationallicence free groups nationallicence Cauchy differences nationallicence finite differences nationallicence Aequationes mathematicae Springer Basel 89/1(2015-02-01), 119-147 0001-9054 89:1<119 2015 89 10 https://doi.org/10.1007/s00010-014-0325-z text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-014-0325-z text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Ng Che Department of Pure Mathematics, University of Waterloo, N2L 3G1, Waterloo, ON, Canada aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/1(2015-02-01), 119-147 0001-9054 89:1<119 2015 89 10