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Zeros of polynomials orthogonal with respect to a signed weight. (English) Zbl 1236.42020
The authors consider the monic orthogonal polynomial sequence \(\{P_n^{\alpha,q} : n\in \mathbb{N}_0\}\) which is orthogonal on \([-1,1]\) with respect to the signed weight \(x^{2q+1}(1 - x^2)^\alpha(1-x)\), \(\alpha > -1\), and \(q\in \mathbb{N}\). They prove that all zeros of these orthogonal polynomials are real, non-interlacing, and that one of the zeros is the endpoint -1.
MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C20 Generalized hypergeometric series, \({}_pF_q\)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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