Linearly-invariant families and generalized Meixner–Pollaczek polynomials

Iwona Naraniecka; Jan Szynal; Anna Tatarczak

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2013)

  • Volume: 67, Issue: 1
  • ISSN: 0365-1029

Abstract

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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 ) = 1 ( 1 - z e i θ ) λ - i x ( 1 - z e i ψ ) λ + i x = ∑ 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  z G λ ( x ; θ , ψ ; z ) is a kernel, will be discussed.

How to cite

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Iwona Naraniecka, Jan Szynal, and Anna Tatarczak. "Linearly-invariant families and generalized Meixner–Pollaczek polynomials." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 67.1 (2013): null. <http://eudml.org/doc/289852>.

@article{IwonaNaraniecka2013,
abstract = {The extremal functions  $f_0(z)$  realizing the maxima of some functionals (e.g. $\max |a_3|$, and  $\max \{arg f^\{^\{\prime \}\}(z)\}$) 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 \[G^\lambda (x;\theta ,\psi ;z)=\frac\{1\}\{(1-ze^\{i\theta \})^\{\lambda -ix\}(1-ze^\{i\psi \})^\{\lambda +ix\}\}=\sum \_\{n=0\}^\infty P\_n^\lambda (x;\theta ,\psi )z^n,\ |z|<1,\] 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.},
author = {Iwona Naraniecka, Jan Szynal, Anna Tatarczak},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {},
language = {eng},
number = {1},
pages = {null},
title = {Linearly-invariant families and generalized Meixner–Pollaczek polynomials},
url = {http://eudml.org/doc/289852},
volume = {67},
year = {2013},
}

TY - JOUR
AU - Iwona Naraniecka
AU - Jan Szynal
AU - Anna Tatarczak
TI - Linearly-invariant families and generalized Meixner–Pollaczek polynomials
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2013
VL - 67
IS - 1
SP - null
AB - The extremal functions  $f_0(z)$  realizing the maxima of some functionals (e.g. $\max |a_3|$, and  $\max {arg f^{^{\prime }}(z)}$) 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 \[G^\lambda (x;\theta ,\psi ;z)=\frac{1}{(1-ze^{i\theta })^{\lambda -ix}(1-ze^{i\psi })^{\lambda +ix}}=\sum _{n=0}^\infty P_n^\lambda (x;\theta ,\psi )z^n,\ |z|<1,\] 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.
LA - eng
KW -
UR - http://eudml.org/doc/289852
ER -

References

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  14. Starkov, V. V., The estimates of coefficients in locally-univalent family U α ' , Vestnik Lenin. Gosud. Univ. 13 (1984), 48–54 (Russian). 
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