  We provide suitable conditions under which the asymptotic limit of the
Hilbert-Samuel coefficients of the Frobenius powers of an
$\mathfrak{m}$-primary ideal exists in a Noetherian local ring
$(R,\mathfrak{m})$ with prime characteristic $p>0.$ This, in turn, gives an
expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also
prove that for a face ring $R$ of a simplicial complex and an ideal $J$
generated by pure powers of the variables, the generalized Hilbert-Kunz
function $\ell(R/(J^{[q]})^k)$ is a polynomial for all $q,k$ and also give an
expression of the generalized Hilbert-Kunz multiplicity of powers of $J$ in
terms of Hilbert-Samuel multiplicity of $J.$ We conclude by giving a
counter-example to a conjecture proposed by I. Smirnov which connects the
stability of an ideal with the asymptotic limit of the first Hilbert
coefficient of the Frobenius power of the ideal.
