Is there any direct relationship exist between p times differentiation of F(z) and f(n) similar to Laplace transform where n times differentiation directly related to its time domain counterpart ?
where F(z) is z-transform of f(n) and p is postive integer
The unilateral z-transform is defined by $$ \tilde{f}(z) = \sum_{n=0}^{\infty} f(n) \, z^{-n}. $$
The derivative is $$ \tilde{f}'(z) = \sum_{n=0}^{\infty} f(n) \, (-n) \, z^{-n-1} = -z^{-1} \sum_{n=0}^{\infty} (nf(n)) \, z^{-n} . $$
Thus, the derivative of the z-transform is (except for the factor $-z^{-1}$) the z-transform of the sequence weighted with "position".
For higher derivatives we get $$ \tilde{f}^{(k)}(z) = (-1)^k z^{-k} \sum_{n=0}^{\infty} (n(n+1)\cdots (n+k-1))f(n)) \, z^{-n} $$