Stability and hyperbolicity

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6. Orthogonal polynomials

Let $(L_n^\alpha)$ denote the generalized Laguerre polynomials. It is known that $L_m^\alpha$ and $L_n^\alpha$ can have common zeros if $|m-n|>1$ . Is the number of common zeros linked to arithmetic properties of $\alpha$ ?

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Conjecture 6.1.
[K. Driver] There are no common zeros between $L_m^\alpha$ and $L_n^\alpha$ , $|m-n|>1$ , for $\alpha$ rational.
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Problem 6.2.
[K. Driver] Let $P_n^{\;\alpha, \,\beta}$ denote the Jacobi polynomial of $\deg n$ with parameters $\alpha, \beta> -1$ . Then $P_n^{\;\alpha', \,\beta'}$ and $P_n^{\;\alpha, \,\beta}$ interlace for $\alpha\approx\alpha'$ , and $\beta\approx\beta'$ . Does the failure of interlacing correspond to some physical law being “violated”?
Add remark
Problem 6.3.
[P. Moussa]
Consider a subspace $S$ of $\mathbb{C}[x]$ of codimension $k$ (defined by $k$ linear constraints, explicitly given). Given a scalar product on $\mathbb{C}[x]$ , find an orthogonal basis of $S$ (ordered by degree); some degrees will be missing.
Which degrees get skipped?
Characterize these orthogonal polynomials.
Does this have anything to do with lacunary Padé approximants?

Cite this as: AimPL: Stability and hyperbolicity, available at http://aimpl.org/hyperbolicpoly.

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