Moreno, Samuel G.; García-Caballero, Esther M. Orthogonality of the Meixner-Pollaczek polynomials beyond Favard’s theorem. (English) Zbl 1270.33005 Bull. Belg. Math. Soc. - Simon Stevin 20, No. 1, 133-143 (2013). 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. Cited in 1 Document 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 Keywords:Meixner-Pollaczek polynomials; Favard’s theorem; non-standard inner product PDFBibTeX XMLCite \textit{S. G. Moreno} and \textit{E. M. García-Caballero}, Bull. Belg. Math. Soc. - Simon Stevin 20, No. 1, 133--143 (2013; Zbl 1270.33005) Full Text: Euclid