Linearly-invariant families and generalized Meixner–Pollaczek polynomials

The extremal functions  $f_0(z)$  realizing the maxima of some functionals (e.g. $max|a_3|$, and  $maxarg{f}^{{}^{\prime }}(z)$) within the so-called universal linearly invariant family $U_{\alpha }$ (in the sense of Pommerenke [10]) have such a form that $f_0^{{}^{\prime }}(z)$  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 $$G^{\lambda }(x;\theta ,\psi ;z)=\frac{1}{(1-z{e}^{i\theta }{)}^{\lambda -ix}(1-z{e}^{i\psi }{)}^{\lambda +ix}}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{\lambda }(x;\theta ,\psi )z^n,|z|<1,$$ 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  $z{G}^{\lambda }(x;\theta ,\psi ;z)$ is a kernel, will be discussed.