Spectrum of Kernel – Discrete orthogonal polynomials

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

K(m,n)=eβ4(m+n+1)22+m+n2m!n!πnm[1Γ(m/2)Γ(n+12)1Γ(n/2)Γ(m+12)]

where m,nN 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

Kk(n,m)=w(m)w(n)fk(n)fk1(m)fk(m)fk1(n)nm.

and one should take the lim 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 \underset{N\rightarrow\infty}{\lim}\ \frac{c^{N}\ \left( \beta\right) _{N}%
}{\Gamma\left( N-x\right) }M_{N}\left( x;\beta,c\right) =\frac{1}{\left(
1-c\right) ^{\beta+x}\Gamma\left( -x\right) }.

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):
K^2(n_1, n_2)= -\frac{2^{3+\frac{n_1+n_2}{2}}}{\sqrt{n_1! n_2 !}} \frac{q^{\frac{1}{2}+\frac{n_1+n_2}{4}}(1-q)^{\frac{n_1+n_2+1}{2}}}{(n_1-n_2)} \times \\
\lim_{N\rightarrow \infty} \frac{(1-q)^{2N}}{\Gamma(N-n_1/2)\Gamma(N-n_2/2)} \left[ M^*_N(\frac{n_1}{2} , \frac{1}{2} , q) \hat M_N(\frac{n_2}{2}, \frac{1}{2} , q) - n_1 \leftrightarrow n_2 \right]

using the asymptotics of the monic Meixner polynomials \widehat{M}_{N}\left( x;\beta,c\right) =\left( \beta\right) _{N}\left(
\frac{c}{c-1}\right) ^{N}\ M_{N}\left( x;\beta,c\right) .

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

\underset{N\rightarrow\infty}{\lim}\frac{\left( c-1\right) ^{N}}
{\Gamma\left( N-x\right) }M_{N}^{\ast}\left( x;\beta,c\right) =-\left(
\frac{c}{1-c}\right) ^{x}\frac{1}{\Gamma\left( -x\right) }B_{c}\left(
-x,1-\beta\right) ,

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?

Answer

Attribution
Source : Link , Question Author : Panagiotis Betzios , Answer Author : Community

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