Bulletin of the Belgian Mathematical Society - Simon Stevin

Orthogonality of the Meixner-Pollaczek polynomials beyond Favard's theorem

Samuel G. Moreno and Esther M. García-Caballero

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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 {\it generalized} ones) share the same three term recurrence relation. The values $\lambda_N=(1-N)/2$, with $N$ a positive integer, 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.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1 (2013), 133-143.

First available in Project Euclid: 18 April 2013

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Primary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

Meixner-Pollaczek polynomials Favard's theorem non-standard inner product


Moreno, Samuel G.; García-Caballero, Esther M. Orthogonality of the Meixner-Pollaczek polynomials beyond Favard's theorem. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 1, 133--143. doi:10.36045/bbms/1366306719. https://projecteuclid.org/euclid.bbms/1366306719

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