Three-term recurrences in Orthogonal polynomials

Question

hi I am reading one lecture note about Orthogonal polynomials (https://www.math.hkbu.edu.hk/ICM/LecturesAndSeminars/08OctMaterials/1/Slide2.pdf) and there's one step in the proof in "Three-term recurrence" section about Favard's theorem I am not really catching.

can someone help explain to me how it is? Thanks!

Answer 1

Equation (7) with $k-1$ instead of $k$ gives \begin{align*} p_{k}(\lambda)-\lambda p_{k-1}(\lambda)=-\alpha_k p_{k-1}(\lambda)-\gamma_{k-1} p_{k-2}(\lambda) +\sum_{j=0}^{k-3}\delta_j p_j(\lambda)\tag{$7^{\prime}$} \end{align*} Taking the inner product of equation ($7^{\prime}$) with $p_k$ from the left gives \begin{align*} \langle p_k,p_k\rangle-\langle p_k, \lambda p_{k-1}\rangle=0\tag{$7^{\prime\prime}$} \end{align*}

We obtain from ($7^{\prime\prime}$) and the definition of the inner product (see formula (3) in OP's referred link): \begin{align*} \color{blue}{\langle\lambda p_k,p_{k-1}\rangle} &=\int_{a}^b\left(\lambda p_k(\lambda)\right)p_{k-1}(\lambda)\,d\alpha(\lambda)\\ &=\int_{a}^bp_{k}(\lambda)\left(\lambda p_{k-1}(\lambda)\right)\,d\alpha(\lambda)\\ &\,\,\color{blue}{=\langle p_k,\lambda p_{k-1}\rangle}\\ &\,\,\color{blue}{=\langle p_{k},p_k\rangle} \end{align*} according to the claim.