For a tridiagonal matrix $T$ with diagonal entries $\alpha_j$ and off-diagonal entries $\beta_j$, Cramer’s rule applied to $(T - z,\mathbb{1})^{-1}$ gives the top-left entry as the closed form

$$[(T - z,\mathbb{1})^{-1}]_{0,0} = \cfrac{1}{(\alpha_0 - z) - \cfrac{\beta_0^2}{(\alpha_1 - z) - \cfrac{\beta_1^2}{(\alpha_2 - z) - \cdots}}}.$$

The cofactor expansion of the relevant determinants produces a two-term recursion in matrix dimension that telescopes into this nested continued fraction. The coefficients of the continued fraction are exactly the entries of $T$.

Combined with the fact that Lanczos produces a tridiagonal projection of any symmetric operator, this is the entire Lanczos-CF correspondence: running Lanczos for $n$ steps on $A$ and evaluating the continued fraction at $z$ gives the resolvent matrix element $\langle v_0 \mid (A - z)^{-1} \mid v_0 \rangle$ as an $[n-1/n]$ Padé approximant, with no separate fitting step.