I have a question about the inverse of lower triangular Toeplitz matrix. There is a matrix $Q$ which can be infinite-dimensional. $ Q=\left[\begin{array}{ccccc} 1\\ -k_{1} & 1\\ -k_{2} & -k_{1} & 1\\ -k_{3} & -k_{2} & -k_{1} & 1\\ \vdots & & & \ddots & \ddots \end{array}\right] $ And the coefficients $0
Now, let $T=Q^{-1}$ is the inverse of this Toeplitz matrix. I read other papers and know $T$ is also lower triangular Teoplitz matrix. Assume the coefficients are $t_i$, and denote $ T=Q^{-1}=\left[\begin{array}{ccccc} 1\\ t_{1} & 1\\ t_{2} & t_{1} & 1\\ t_{3} & t_{2} & t_{1} & 1\\ \vdots & & & & \ddots \end{array}\right] $
I want to know whether $\{t_i\}$ are convergent, and moreover, will $\frac{t_{i+1}}{t_i}$ converge to a constant when $i$ is large enough ($i\rightarrow\infty$)? I tried some examples, and the results showed that both of them are convergent to a constant. But I don't know how to prove it.
I hope to discuss with you. Thanks in advance.
[Updated]
Considering @mike's idea, now I am thinking whether we can express the question as the coefficient ratio for a power series.
As $Q=L^{0}+\sum_{i=1}^{\infty}\left(-k_{i}L^{i}\right)$, where $L$ is the shift matrix in which $L_{ij}=\delta_{i,j+1}$ in infinite-dimensional ($L^0=I$). And $T$ can be written as $T=L^{0}+\sum_{i=1}^{\infty}\left(t_{i}L^{i}\right)$.
So the question is whether ${t_i}$ is convergent and whether $\lim_{i\rightarrow\infty}\frac{t_{i+1}}{t_i}$ is a constant ($z_0$). If yes, what's the expression of $z_0$.
Is this related to the power-series? I don't have much knowledge about it...