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Orthogonality of the Meixner-Pollaczek polynomials beyond Favard’s theorem. (English) Zbl 1270.33005

Summary: We extend the family of Meixner-Pollaczek polynomials \(\{P_n^{(\lambda)}(\cdot;\phi)\}_{n=0}^{\infty}\), classically defined for \(\lambda>0\) and \(0<\phi<\pi\), to arbitrary complex values of the parameter \(\lambda\) in such a way that both polynomial systems (the classical and the new generalized ones) share the same three-term recurrence relation. The values \(\lambda_N=(1-N)/2\), for a positive integer \(N\), are the only ones for which no orthogonality condition can be deduced from Favard’s theorem.
In this paper, we introduce a non-standard discrete-continuous inner product with respect to which the generalized Meixner-Pollaczek polynomials \(\{P_n^{(\lambda_N)}(\cdot;\phi)\}_{n=0}^{\infty}\) become orthogonal.

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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Full Text: Euclid