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

$$K(m,n)= e^{-\frac{\beta}{4} (m+n+1)} \frac{2^{2+\frac{m+n}{2}}}{\sqrt{m! n!}} \frac{\sqrt{\pi}}{n-m} \left[ \frac{1}{\Gamma(-m/2)\Gamma(\frac{-n+1}{2})} - \frac{1}{\Gamma(-n/2)\Gamma(\frac{-m+1}{2})} \right]$$

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

$$K_k(n,m)= \sqrt{w(m) w(n)} \frac{f_k(n) f_{k-1}(m)- f_k(m) f_{k-1}(n)}{n-m}. $$

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

  • $\begingroup$ Added discussion $\endgroup$ Jun 28, 2016 at 18:00


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy