## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

#### Abstract

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

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

Dates
First available in Project Euclid: 18 April 2013

https://projecteuclid.org/euclid.bbms/1366306719

Digital Object Identifier
doi:10.36045/bbms/1366306719

Mathematical Reviews number (MathSciNet)
MR3082748

Zentralblatt MATH identifier
1270.33005

#### Citation

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