This paper proves that a generalized Hertz pressure (the product of Hertz square root and an even polynomial of degree $2n$ with respect to coordinates) applied over elastic half-space boundary generates a polynomial normal displacement of degree $2n+2$. Polynomial surface coefficients are combinations of elliptical integrals. The equation of rigid punch surface generating this pressure is derived, as well as the conditions in which an elliptical contact occurs. For second order surfaces, $n=0$, these results yield all Hertz formulas, whereas new formulas are derived for contact parameters between fourth, sixth, and eight order surfaces.

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