The Meixner--Pollaczek polynomials and a system of orthogonal polynomials in a strip
Two systems of polynomials \(\{\tau_n\}\) and \(\{\sigma_n\}\) which satisfy the recursion relations \[ \tau_{n+1}(z)=z\tau_n(z)-n^2\tau_{n-1}(z), \quad n\geq0, \qquad \tau_{-1}=0, \;\tau_0=1 \] and \[ \sigma_{n+1}(z)=z\sigma_n(z)-n(n-1)\sigma_{n-1}(z), \quad n\geq0, \qquad \sigma_{-1}=0, \;\sigma_0=1, \] respectively, are under consideration in the present paper. The first system is known to be a special case of the symmetric Meixner-Pollaczek polynomials orthogonal with respect to the weight function \(w=1/2\cosh(\pi x/2)\). The second one turns out to form an orthogonal basis in the Hilbert space of functions analytic in a certain strip of the complex plane. The author finds some simple operators that connect the systems with each other which makes it possible to study the asymptotic behavior of the second system.
scientific article; zbMATH DE number 2085567
scientific article