Linearly-invariant families and generalized Meixner–Pollaczek polynomials

Iwona Naraniecka; Jan Szynal; Anna Tatarczak

Iwona Naraniecka ; Jan Szynal ; Anna Tatarczak

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 67 (2013), / Harvested from The Polish Digital Mathematics Library

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Résumé

The extremal functions $f_{0} (z)$ realizing the maxima of some functionals (e.g. $max | a_{3} |$ , and $max a r g f^{^{'}} (z)$ ) 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 $G^{λ} (x; θ, ψ; z) = \frac{1}{{(1 - z e^{i θ})}^{λ - i x} {(1 - z e^{i ψ})}^{λ + i x}} = \sum_{n = 0}^{\infty} P_{n}^{λ} (x; θ, ψ) z^{n}, | z | < 1,$ where the parameters $λ$ , $θ$ , $ψ$ satisfy the conditions: $λ > 0$ , $θ \in (0, π)$ , $ψ \in ℝ$ . 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 $z G^{λ} (x; θ, ψ; z)$ is a kernel, will be discussed.

Publié le : 2013-01-01
EUDML-ID : urn:eudml:doc:289852

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     title = {Linearly-invariant families and generalized Meixner--Pollaczek polynomials},
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     volume = {67},
     year = {2013},
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Iwona Naraniecka; Jan Szynal; Anna Tatarczak. Linearly-invariant families and generalized Meixner–Pollaczek polynomials. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 67 (2013) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2013_67_1_45-56/

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