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Bispectral Jacobi type polynomials. (English) Zbl 1487.42062

The authors study the bispectrality of Jacobi type polynomials, which are eigenfunctions of higher-order differential operators and can be defined by taking suitable linear combinations of a fixed number of consecutive Jacobi polynomials. Jacobi type polynomials include, as particular cases, the Krall-Jacobi polynomials. As the main results the authors prove that the Jacobi type polynomials always satisfy higher-order recurrence relations (i.e., they are bispectral). This paper also provides a proof concerning that the Krall-Jacobi families are the only Jacobi type polynomials which are orthogonal with respect to a measure on the real line.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33E30 Other functions coming from differential, difference and integral equations
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References:

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