On the Hilbert coefficients, depth of associated graded rings and reduction numbers

Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J=(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_d)$ a minimal reduction of $I$. We show that if $J_{d-1}=(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_{d-1})$ and $\sum\limits_{n=1}^\infty\lambda{({I^{n+1}\cap J_{d-1}})/({J{I^n} \cap J_{d-1}})=i}$wherei=0,1,thendepth$G(I)\geq{d-i-1}$. Moreover, we prove that if $e_2(I)={\sum }_{n=2}^{\mathrm{\infty }}(n-1)\lambda (I^n\mathrm{/}J{I}^{n-1})-2;$ or if $I$ is integrally closed and $e_2(I) = \sum_{n=2}^\infty (n-1)\lambda({{I^{n}}}/JI^{n-1})-i$where$i=3,4$,then$e_1(I) = \sum_{n=1}^\infty \lambda(I^n / JI^{n-1})-1.$Inaddition,weshowthat$r(I)$isindependent\mathrm{.}Furthermore,westudytheindependenceof$r(I)$ with some other conditions.