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Given a commutative local ring $(R,\mathfrak m)$ and an ideal $I$ of $R$, afamily of quotients of the Rees algebra $R[It]$ has been recently studied as aunified approach to the Nagata's idealization and the amalgamated duplicationand as a way to construct interesting examples, especially integral domains.When $R$ is noetherian of prime characteristic, we compute the Hilbert-Kunzfunction of the members of this family and, provided that either $I$ is$\mathfrak{m}$-primary or $R$ is regular and F-finite, we also find theirHilbert-Kunz multiplicity. Some consequences and examples are explored.

The Hilbert-Kunz function of some quadratic quotients of the Rees algebra