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.