Linearly-invariant families and generalized Meixner–Pollaczek polynomials

Abstract

The extremal functions  \(f_0(z)\)  realizing the maxima of some functionals (e.g. \(\max|a_3|\), and  \(\max{arg f^{'}(z)}\)) within the so-called universal linearly invariant family \(U_\alpha\) (in the sense of Pommerenke [10]) have such a form that \(f_0^{'}(z)\)  looks similar to generating function for Meixner-Pollaczek (MP) polynomials [2], [8]. This fact gives motivation for the definition and study of the generalized Meixner-Pollaczek (GMP) polynomials \(P_n^\lambda(x;\theta,\psi)\) of a real variable \(x\) as coefficients of \[G^\lambda(x;\theta,\psi;z)=\frac{1}{(1-ze^{i\theta})^{\lambda-ix}(1-ze^{i\psi})^{\lambda+ix}}=\sum_{n=0}^\infty P_n^\lambda (x;\theta,\psi)z^n,\ |z|<1,\] where the parameters \(\lambda\), \(\theta\), \(\psi\) satisfy the conditions: \(\lambda > 0\), \(\theta \in (0,\pi)\), \(\psi \in \mathbb{R}\). In the case \(\psi=-\theta\) we have the well-known (MP) polynomials. The cases \(\psi=\pi-\theta\) and \(\psi=\pi+\theta\) leads to new sets of polynomials which we call quasi-Meixner-Pollaczek polynomials and strongly symmetric Meixner-Pollaczek polynomials. If  \(x=0\),  then we have an obvious generalization of the Gegenbauer polynomials.

The properties of (GMP) polynomials as well as of some families of holomorphic functions  \(|z|<1\)  defined by the Stieltjes-integral formula, where the function  \(zG^{\lambda}(x; \theta, \psi;z)\) is a kernel, will be discussed.