Geodesic
Linearly-invariant families and generalized Meixner–Pollaczek polynomials
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 67 (2013) no. 1.

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The extremal functions  f_0(z)  realizing the maxima of some functionals (e.g. max|a_3|, and  maxarg 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-ze^iθ)^λ-ix(1-ze^iψ)^λ+ix=∑_n=0^∞ P_n^λ (x;θ,ψ)z^n, |z| lt;1, where the parameters λ, θ, ψ satisfy the conditions: λ gt; 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| lt;1  defined by the Stieltjes-integral formula, where the function  zG^λ(x; θ, ψ;z) is a kernel, will be discussed. 
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Naraniecka, Iwona; Szynal, Jan; Tatarczak, Anna. Linearly-invariant families and generalized Meixner–Pollaczek polynomials. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 67 (2013) no. 1. http://geodesic.mathdoc.fr/item/AUM_2013_67_1_a2/

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