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6. Orthogonal polynomials

    1.     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 $\alpha$?

      Conjecture 6.1.

      [K. Driver] There are no common zeros between $L_m^\alpha$ and $L_n^\alpha$, $|m-n|>1$, for $\alpha$ rational.
        • 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”?
            • 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.
              1. Which degrees get skipped?
              2. Characterize these orthogonal polynomials.
              3. Does this have anything to do with lacunary Padé approximants?

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