Orthogonalizing polynomials with inner product depending on parameters

Question

I need to orthogonalize the polynomials $h_n(x)=x^{2n}(1+x^2)^{-4S}$ with $x\in\textbf{R}$, $2S\in\textbf{N}$ and $n\in\{1,3,5,\ldots, 4S-1\}$ over the inner product

$\langle h_n,h_m\rangle=32\pi S\displaystyle\int_0^1h_n(x)h_m(x)\frac{x(1-x^2)dx}{(1+x^2)^3}$

$=16\pi S\frac{\displaystyle\sum_{k=0}^{8S-n-m}{8S+2 \choose k}(8S+1-n-m-k)}{\displaystyle2^{8S+1}(n+m+2)(n+m+1){8S+2 \choose 8S-n-m}}$ .

Using the integral itself with Orthogonalize takes forever and the result is not informative. So I'd like to use the formula above in terms of the parameters $n$ and $m$ instead. Is there a way to do this with Mathematica, i.e. use Orthogonalize with an inner product that depends on parameters rather than the functions and their variables?

Answer 1

Define your scalar product of the h polynomials:

p[h[n_], h[m_]] := 16 Pi S Sum[Binomial[8 S + 2, k] (8 S + 1 - n - m - k), {k, 0, 8 S - n - m}] / (2^(8 S + 1) (n + m + 2) (n + m + 1) Binomial[8 S + 2, 8 S - n - m])

Add these properties:

p[c_ pol1_h, pol2_] := c p[pol1, pol2]
p[pol1_, c_ pol2_h] := c p[pol1, pol2]

p[sum_Plus, pol_] := p[#, pol] & /@ sum
p[pol_, sum_Plus] := p[pol, #] & /@ sum

p[c_ sum_Plus, pol_] := p[Expand[c sum], pol]
p[pol_, c_ sum_Plus] := p[pol, Expand[c sum]]

Take the case

S = 2;
hs = {h[1], h[3], h[5], h[7]};

Then I think this is what you want:

ohs = Expand@ Orthogonalize[hs, p]

Check orthonormality with

Outer[p, ohs, ohs] // FullSimplify