Rational-form conjecture for the central recurrence coefficient

Let $X(\eta)=I[\Xi_{\mathcal{D}}Y](\eta)$ have degree $L$, and let $r_{n,k}^{X,\mathcal{D}}$ be the coefficients in the $1+2L$-term constant-coefficient recurrence for the multi-indexed Meixner–Pollaczek or continuous Hahn polynomials. Let $\mathcal{E}_n$ denote the spectral value and define $\alpha_j(z)$ for $1\leq j\leq 2L$ as in the paper. Central-coefficient rational-form conjecture. As a function of $n$, the coefficient $r_{n,0}^{X,\mathcal{D}}$ has the form

$$r_{n,0}^{X,\mathcal{D}}=-\left.\frac{I(z)}{\prod_{j=1}^{L}\alpha_j(z)\alpha_{2L+1-j}(z)}\right|_{z=\mathcal{E}_n},$$

where $I(z)$ is a polynomial of degree $L$ for Meixner–Pollaczek type and of degree at most $2L$ for continuous Hahn type. This conjecture gives an explicit spectral-variable description of the central recurrence coefficient; it is presented without a proof in the supplied text.