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Proving an identity related related to Stieltjes transformation and continued fractions

This question is reposted from Math Stack Exchange (you can see the original post here). The motivation for reposting is that I feel like the question isn't getting much attention in MSE - if there is something wrong with reposting I have no problem with deleting the question and obviously being sorry for any inconvenience caused. Follows the question:

Conext and notation. In the question below, $R_n(z) / S_n(z)$ denotes the n-th convergent of the continued fraction

$$ \frac{1|}{|z-b_1} + \frac{-a_2|}{|z-b_2} + \dots + \frac{-a_n|}{|z-b_n} + \dots,$$

where $a_n,b_n$ are the coefficients of a generic orthogonal sequence of monic polynomials $\{p_n(z)\}$ that satisfies the three term recurrence relation

$$ p_n(z) = (x-b_n)p_{n-1}(z) - a_np_{n-2}(z), \quad \text{ for } \, \, n= 1,2,\dots$$

The problem. Define the function

$$ \hat w(z) = \int_a^b \frac{1}{z-t}w(t) dt,$$ which is normally known as the Stieltjes transformation. I wish to prove that

\begin{equation} \tag{1} \hat w(z) - \frac{R_n(z)}{S_n(z)} = \frac{1}{p_n(z)}\int_a^b \frac{p_n(t)}{z-t}w(t) \, dt.\end{equation}

and

\begin{equation} \tag{2} \hat w(z) - \frac{R_n(z)}{S_n(z)} = \frac{k_n^1}{z^{2n+1}} + \frac{k_n^2}{z^{2n+2}} + \dots, \quad |z| > R. \end{equation}

My attempt. I was able to prove $(1)$ with ease. Identity $(2)$ gave me quite some more problems. Follows my attempt:

We have that

\begin{equation*} \begin{split} \hat w(z) - \frac{R_n(z)}{S_n(z)} &= \frac{1}{p_n(z)} \int_a^b \frac{p_n(t)}{z-t}w(t) \, dt \\[.25cm] &= \frac{1}{p_n(z)} \int_a^b \sum_{j=0}^\infty \frac{t^j}{z^{j+1}}p_n(t) w(t) \, dt \\[.25cm] &= \frac{1}{p_n(z)} \sum_{j=0}^\infty \frac{1}{z^{j+1}} \boxed{\int_a^b t^j p_n(t) w(t) \, dt}. \end{split} \end{equation*}

So, all we have to do is study the boxed integral above. From the theory of orthogonal polynomials, we know that for $j<n$ this integral is zero and for $j=n$ we have that

$$ \int_a^b t^np_n(t) w(t) \, dt = \gamma_n h_n, $$

where $\displaystyle{h_n = \int_a^b p_n^2(t)w(t) \, dt}$ and $\gamma_n$ is such that $\displaystyle{t^n = \sum_{i=0}^n \gamma_i p_i(t)}$ (recall that $\{p_0(t),\dots,p_n(t)\}$ forms a basis for the vectorial space of the polynomials in one variable of degree equal or smaller than $n$). So everything we have to do is to study the boxed integral for values of $j$ such that $j < n.$ For this cases, it is clear that $\{ p_0(t),\dots,p_j(t)\}$ forms a basis for the vectorial space of the polynomials in one variable of degree equal or smaller than $j$. Therefore, we can find scalars $\delta_i$ such that $$ t^j = \sum_{i=0}^j \delta_i p_i(t). $$ Then, $$ \int_a^b t^j p_n(t) w(t) \, dt = \sum_{i=0}^j \delta_i \int_a^b p_i(t)p_n(t) w(t) \, dt = \delta_n h_n. $$ Therefore, we have that

$$ \hat w(z) - \frac{R_n(z)}{S_n(z)} = \frac{1}{p_n(z)}\left[ \frac{\gamma_n h_n}{z^{n+1}} + \sum_{j=n+1}^\infty \frac{\delta_n h_n}{z^{j+1}} \right].$$

I can see some similarities with the result we wish to prove but at the same time I think I am quite far away. I don't know how to simplify this further, thought.

Thanks for any help in advance.