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We derive explicit representation formulae of Green functions for GJMSoperators on $n$-spheres, including the fractional ones. These formulae notonly have natural geometric interpretations concerning the extrinsic geometryof the round sphere, but also reflect the spherical rigidity among closedembedded hypersurfaces in $\mathbb{R}^{n+1}$.

Green functions for GJMS operators on spheres, Gegenbauer polynomials and rigidity theorems