How to solve this generic linear equation system?

Suposse I have to solve this system:

$$\left( \begin{array}{cccc} r_g(0) & r_g(1) & \cdots & r_g(N-1) \\ r_g(1) & r_g(0) & \cdots & r_g(N-2) \\ \vdots & \vdots & \ddots & \vdots \\ r_g(N-1) & r_g(N-2) & \cdots & r_g(0) \end{array} \right) \left( \begin{array}{c} h(0) \\ h(1)\\ \vdots \\ h(N-1) \end{array} \right) = \left( \begin{array}{c} g(0) \\ 0\\ \vdots \\ 0 \end{array} \right)$$

Where the unkown variables are $$h(0), h(1), \cdots, h(N-1)$$. Moreover, $$g(n)$$ is:

$$g(n) = \alpha^{n+1}u(n) - \alpha^{n-1}u(n-1) \quad \text{with } \left \lbrace \begin{array}{c} |\alpha| < 1 \\ n = 0,1,\cdots,\infty \end{array} \right.$$

$$u(n)$$ is a function that gives $$1$$ for $$n \geq 0$$ and $$0$$ for $$n < 0$$. On the other hand, the definition of $$r_g(n)$$ is:

$$r_g(k) = \sum_{n=0}^{\infty}g(n)g(n-k)$$

If I didn't make any mistake, this results:

$$r_g(k) = \frac{\alpha^k (\alpha^2+1)^2}{1-\alpha^2}$$

Thus, the system I have to solve is:

$$\frac{(\alpha^2+1)^2}{1-\alpha^2} \left( \begin{array}{cccc} 1 & \alpha & \cdots & \alpha^{N-1} \\ \alpha & 1 & \cdots & \alpha^{N-2} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha^{N-1} & \alpha^{N-2} & \cdots & 1 \end{array} \right) \left( \begin{array}{c} h(0) \\ h(1)\\ \vdots \\ h(N-1) \end{array} \right) = \left( \begin{array}{c} \alpha \\ 0\\ \vdots \\ 0 \end{array} \right)$$

The problem is that I need an analytical expression for $$h(n)$$ but is a $$NxN$$ matrix where the coefficient $$N$$ is unknown.

I hope someone can help me. Thank you for your answers.