Let $L: \mathbb{R}[x] \rightarrow \mathbb{R}$ be a positive definite linear functional and let that $\{s_n\}$ be a positive semi-definite sequence such that $$L(x^n)= s_n, n\ge 0$$ and $$<p,q> = L(pq).$$ Given a positive definite sequence, we use the Gram-Schimdt orthogonalization method to construct a sequence of orthogonal polynomials $\{p_n\}$ whose leading coefficient is positive due to the positivity nature of the sequence given. It turns out that this sequence of orthogonal polynomials $\{p_n\}$ satsifies a three term recurrence relation given below \begin{equation} xp_n(x) =b_np_{n+1}(x)+a_np_n(x)+b_{n-1}p_{n-1}(x) , \quad n\ge 0 \end{equation}
We can see the sequence $p_n(x)$ as a solution to the three term recurrence relations stated above. Akhiezer http://www.maths.ed.ac.uk/~aar/papers/akhiezer.pdf as my reference introduced another solution to this three term recurrence relation by defining another solution by \begin{equation} q_n(x)= \displaystyle L\left(\frac{p_n(x)-p_n(y)}{x-y}\right) \end{equation} where the quotient $\frac{p_n(x)-p_n(y)}{x-y}$ is a polynomial in $x$ and $y$ and $q_n(x)$ is a polynomial in variable $x$ and its degree is $n-1$ for any $x,y \in \mathbb{R}$ so that we have $ \displaystyle xq_n(x) =b_nq_{n+1}(x)+a_nq_n(x)+b_{n-1}q_{n-1}(x), n\ge 1$ with $q_0(x)=0$ and $q_1(x)= \frac{1}{b_0}$
Question: Akhiezer claimed that this sequence of polynomials $\{q_n\}$ is orthogonal. I dont understand how this is true. Can anyone please show me what this true?
PS: Recall that $$<p,q> = L(pq)$$ defines an inner product
In fact what is meant, is that the sequence $(q_n)_{n\ge0}$ is orthogonal with respect to another positive definite linear functional $\tilde{L}$ (i.e. such that $\tilde{L}(q_n,q_m)=0$ for $ n\ne m$). This, in fact, follows from a theorem called the Shohat–Favard theorem which roughly says that a sequence of polynomials $(q_n)_{n\ge0}$ that satisfy a $3$ terms recurrence relation of the form $$b_{n+1}q_{n+1}(x)= (x-a_n)q_n(x)-b_{n-1}$$ with the first nonzero term being constant, and such that the $a_n$'s are real, and the $b_n$'s are positive, must be orthogonal for some positive linear function $\tilde{L}$.
For more details one may consult this and the references therein.