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 \(max{arg f'(z)}\)) within the so-called universal linearly invariant family \(U_α\) (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 \(P_n^λ(x;θ,ψ)\) of a real variable \(x\) as coefficients of \[G^λ(x;θ,ψ;z)={1}{(1-zeiθ)λ-ix(1-zeiψ)λ+ix}=∑ₙ₌₀^∞ P_n^λ (x;θ,ψ)z^n,\ |z|<1,\] where the parameters \(λ\), \(θ\), \(ψ\) satisfy the conditions: \(λ > 0\), \(θ ∈ (0,π)\), \(ψ ∈ R\). In the case \(ψ=-θ\) we have the well-known (MP) polynomials. The cases \(ψ=π-θ\) and \(ψ=π+θ\) 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 \(zGλ(x; θ, ψ;z)\) is a kernel, will be discussed.