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 $J=(x_1,...,x_d)$ a minimal reduction of $I$. We show that if $J_{d-1}=(x_1,...,x_{d-1})$ and $\sum\limits_{n=1}^\infty\lambda{({I^{n+1}\cap J_{d-1}})/({J{I^n} \cap J_{d-1}})=i}$ where i=0,1, then depth $G(I)\geq{d-i-1}$. Moreover, we prove that if $e_2(I) = \sum_{n=2}^\infty (n-1) \lambda (I^n/JI^{n-1})-2;$ 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 $e_1(I) = \sum_{n=1}^\infty \lambda(I^n / JI^{n-1})-1.$ In addition, we show that $r(I)$ is independent. Furthermore, we study the independence of $r(I)$ with some other conditions.