I am reading Atkinson's "An Introduction to Numerical Analysis" and I am trying to understand how a certain equation was reached. It is called the "Triple Recursion Relation" for an orthogonal family of polynomials, from page 214.
My difficulty, which may or may not lead to the solution is: how do I calculate $(x\phi_n,\phi_n)$? The context is within $ d_n = -a_n \frac{\int_a^b x \phi_n \phi_n w(x) \ \mathrm dx}{\int_a^b \phi_n \phi_n w(x) \ \mathrm dx}$ (This is a different approach than the approach used in the link below).
Relevant information:
The relation is: $ \phi_{n+1}(x) = (a_n x + b_n) \phi_n(x) - c_n \phi_{n-1}(x)$ It is our goal to calculate the terms $b_n$ and $c_n$. I think $b_n = d_n$ and $c_n = d_{n-1}$, by comparing equations (4.4.21) and (4.4.24).
$\phi_n(x)$ are polynomials from an orthogonal family with weight function $w(x)$. The interval of interest is $(a,b)$. That means: $ (\phi_i, \phi_j) = \int_a^b w(x) \phi_i(x) \phi_j(x) \ \mathrm dx = 0, \text{if } i \neq j$
Also, $ (\phi_i, \phi_j) \neq 0, \text{if } i = j$
There is a representation of $\phi_n(x)$, given as: $\phi_n(x) = A_n x^n + B_n x^{n-1} + \ldots = A_n(x - x_{n,1})(x - x_{n,2}) \ldots (x-x_{n,n})$. Also, $ a_n = \frac{A_{n+1}}{A_n}$ It is also stated $\gamma_n = (\phi_n, \phi_n) \geq 0$.
The (solution) coefficients are given as $ b_n = a_n \left[ \frac{B_{n+1}}{A_{n+1}} - \frac{B_n}{A_n} \right]$ and $ c_n = \frac{A_{n+1} A_{n-1}}{A_n^2} \cdot \frac{\gamma_n}{\gamma_{n-1}}$
$G(x)$ is defined as: $ G(x) = \phi_{n+1}(x) - a_n x \phi_n(x) = \left[ B_{n+1} - \frac{A_{n+1}B_n}{A_n} \right] x^n + \ldots = d_n \phi_n(x) + d_{n-1} \phi_{n-1}(x)$
The equation for $d_i$ is also given: $ d_i = \frac{1}{\gamma_i} \left[ (\phi_{n+1}, \phi_i) - a_n(x\phi_n, \phi_i)\right]$
Note that I also found a external text which shows a proof of how the coefficients are reached. However, I do not understand one of the lines, and also I think the exact solution is a little bit different (it considers orthonormal vs. our orthogonal polynomials). The text is located here (page 64 on the page label). The exact equation there that I do not understand is: $ x P_{n-1} = \frac{\alpha_{n-1}}{\alpha_n} P_n + q_{n-1}$ Converting to our notation, the equation is $ x \phi_{n-1}(x) = \frac{d_{n-1}}{d_n} \phi_n + G_{n-1}(x)$
I am not sure what the precise meaning of $G_{n-1} (x)$ is.
I think all of the information I need is here, although I may be missing a "connection" step. Can someone give me a hint or some kind of "strategy" approach to solving this type of problem? I can provide more details if necessary.
Thanks.