Finding general orthogonal polynomials 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???
 A: So it seems you want to know what can be said about the zeroes of your polynomials for $k\neq 1$, since for $k=1$ we have classical Laguerre polynomials. I don't have the answer for you but this might lead you in the right direction.
The way you should consider your problem is as a deformation of the Laguerre orthogonality measure
$$ \langle f,g \rangle_k = \int_{0}^{\infty} f(x) g(x) e^{-t x} e^{-x} dx, $$
where $t=k-1$ from your example.
This deformation is called the Toda deformation,
because if you consider polynomials that are orthonormal to this measure, than their three-term-recurrence coefficients satisfy the Toda equations in Flaschka coordinates.
Three term recurrence:
$$
x p_n (x,t) = a_{n+1}(t) p_{n+1}(x,t) + b_n (t) p_n (x,t) + a_n (t) p_{n-1} (x,t)
$$
and the Toda equations:
$$
\frac{d}{dt} a_{n}^2 = a_{n}^2 \left( b_{n-1} - b_{n} \right), \quad n\geq 1,
$$
$$
\frac{d}{dt} b_n = a_{n}^2-a_{n+1}^2, \quad n\geq 0.
$$
For proofs of this fact see theorem 2.8.1 in Classical and Quantum Orthogonal Polynomials in One Variable, M. E. H. Ismail or theorem 3.8 in Orthogonal Polynomials and Painlevé Equations, W. Van Assche.
From coefficients $a_n,b_n$ you can recover the polynomial coefficients and from those the zeroes.
