On some positive definite functions

We study the function $(1-\mathrm{\parallel }x\mathrm{\parallel })\text{slash}(1-\mathrm{\parallel }x{\mathrm{\parallel }}^r),$ and its reciprocal, on the Euclidean space ${\mathbb{R}}^n,$ with respect to properties like being positive definite, conditionally positive definite, and infinitely divisible.