Polynomials of the second kind of almost "the second kind" again

Question

Let $s=(s_n)_{n\geq 0}$ be a positive definite sequence with orthonormal polynomials $(p_n)_{n\geq 0}$ and polynomials $(q_n)_{n\geq 0}$ of the second kind. Then $p^*_n=q_{n+1}$, $n\geq 0$ are the orthonormal polynomials with respect to some positive definite sequence $s^*$ (through the Farvard's Theorem). How do you show that the orresponding polynomials $q^*_n$ of the second kind are given by $$ q^*_n(x)=\frac{p_1(x)}{q_1(x)}q_{n+1}-p_{n+1}(x)\quad ? $$ Recall that $$ q_n(x)=L_{s,y}\left ( \frac{p_n(x)-p_n(y)}{x-y} \right ) $$ where $L_{s,y}$ is the linear function that acts on the variable $y$ defined by $L_{s,y}(y^n)=s_n$.