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Some Laguerre-Hahn orthogonal polynomials of class one. (English) Zbl 1415.33004

Summary: Let \(u(\beta_0)\) be the regular form fulfilling \((u(\beta_0))_{2n+1}=\beta_0(u(\beta_0))_{2n}\), \(n\geq 0\) where \(\beta_0\) is an arbitrary complex number in such a way that for \(\beta_0=0\) one has the symmetric forms. Recently, the symmetric Laguerre-Hahn forms (when \(\beta_0=0\)) of class \(s\leq1\) are determined. In this paper, we determine all the Laguerre-Hahn forms \(u(\beta_0)\) of class \(s=1\), when \(\beta_0\neq0\), through the resolution of a nonlinear system satisfied by the coefficients of the three-term recurrence relation of their sequences of monic corresponding orthogonal polynomials.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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References:

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