# Spectrum of Kernel – Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form

where $m, n \in \mathbb{N}$ and $0 < q<1$ a real parameter.

I want to diagonalize this kernel and find its eigenvalues/eigenfunctions.

It looks like an integrable kernel after having performed the Cristoffel-Darboux summation formula which has a general form like

and one should take the $\lim_{k \rightarrow \infty}$ in this kernel to match with the previous one.

This indicates a possible solution if I find discrete polynomials $f_k (n)$ such that $\lim_{k \rightarrow \infty} f_k(n) \sim \Gamma(-n/2)$.

One should also satisfy $f_{k-1}(n) \sim f_k(n-1)$ to account for the shift in the Gamma functions.

Is there a solution to this problem? Do such polynomials exist?

Edit: Based on the paper http://arxiv.org/pdf/1406.6193.pdf , one could try to use asymptotic expansions for Meixner polynomials and in particular the relation
that holds for all complex numbers $x$.
A fact that makes the identification hard is the $\Gamma(N-x)$ in the denominator...

Since I am actually interested in the square of the operator $\hat K$, I would also be interested to see if one could maybe perform this identification with some Meixner kernel at least for the square of the operator that I managed to bring into the following form (note $n_1$, $n_2$ are odd):

using the asymptotics of the monic Meixner polynomials

and the following asymptotic form of the associated Meixner polynomials $M^*_N$ (these have shift 1 in the recursion relation):

where $B_{z}\left( a,b\right)$ is the incomplete Beta function.

Is now $\hat K^2$ a Christoffel-Darboux kernel? Can I write it in a form like $\sim \sum_{k=0}^\infty \hat M_k M^*_k$? What about its spectrum?