Many Special functions are orthogonal; for example, the sine and cosine function is an orthogonal function. Also, a couple of orthogonal polynomials are well-known. Now I'm asking the following: Given that the $n$-th orthogonal polynomial $p_n(x)$ (it is multiplied by a damping factor that is necessary for integral convergence) can be represented as a product (because it has $n$ zeroes):
$p_n(x)= a_{0,n} e^{-kx/2} \prod_{i=1}^n (x-a_{i,n})$.
What relation must hold between the Parameters $a_{i,n}$ if it must hold $\int_0^\infty p_n(x)p_m(x)dx = \delta_{mn}$?
My thoughts:
The constants $a_{0,n}$ can be determined only from the normalization condition. For finite $n$ I can expand out the product and use the Gamma function; then I know that it arises a System of quadratical equations in order to fit to the orthogonality condition. But if I want to construct infinitely many orthogonal polynomials, then the System of quadratic equations is non-solvable. How can I construct orthogonal polynomials in this case???