# Orthogonal polynomials of the second kind

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 $L(x^n)= s_n, n\ge 0.$ Given a positive definite sequence, I was able to 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 $$xp_n(x) =b_np_{n+1}(x)+a_np_n(x)+b_{n-1}p_{n-1}(x) , \quad n\ge 0$$

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 $$q_n(x)= \displaystyle L\left(\frac{p_n(x)-p_n(y)}{x-y}\right)$$ 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 $q_1(x)= \frac{1}{b_0}$

My question is how can the linear functional $L$ be defined on rational functions since its domain is $\mathbb{R}[x]$. I have troubles understanding this definition even though I actually confirmed the definition is true by computing $q_1,q_2$.You can please check Akhiezer page 8 for more clarification. Details and explanation will be much appreciated.

It seems like the term in brackets in the left $$q_n(x)= \displaystyle L\left(\frac{p_n(x)-p_n(y)}{x-y}\right)$$ is actually a polynomial due to the Taylor series, just replace $$p_n(x)$$ by its Taylor series at y.