The Meixner–Pollaczek polynomials and a system of orthogonal polynomials in a strip

Abstract

Considered in this paper are two systems of polynomials that are orthogonal systems for two different but related inner product spaces. One of these systems is a special case (λ= 1 2 ) of the symmetric Meixner–Pollaczek polynomial systems, P n ( λ) ( x/2, π/2), and it turns out that this system is closely related to a system of orthogonal polynomials in the strip, S={ z:−1<Im( z)<1}. Moreover, there are some simple operators that connect the systems with each other. We have designated the special case of the symmetric Meixner–Pollaczek polynomial systems by τ ̃ n and the latter system on the strip by σ ̃ n , and we have been able to show that this system is the limiting case of the symmetric Meixner–Pollaczek polynomial systems, P n ( λ) ( x/2, π/2) as λ→0.