# Linearly-invariant families and generalized Meixner–Pollaczek polynomials

Iwona Naraniecka; Jan Szynal; Anna Tatarczak

Annales UMCS, Mathematica (2013)

- Volume: 67, Issue: 1, page 45-56
- ISSN: 2083-7402

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topIwona Naraniecka, Jan Szynal, and Anna Tatarczak. "Linearly-invariant families and generalized Meixner–Pollaczek polynomials." Annales UMCS, Mathematica 67.1 (2013): 45-56. <http://eudml.org/doc/268070>.

@article{IwonaNaraniecka2013,

abstract = {The extremal functions f0(z) realizing the maxima of some functionals (e.g. max |a3|, and max arg f′(z)) within the so-called universal linearly invariant family Uα (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 Pλn (x; θ,ψ) of a real variable x as coefficients of [###] 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.},

author = {Iwona Naraniecka, Jan Szynal, Anna Tatarczak},

journal = {Annales UMCS, Mathematica},

keywords = {Typically-real functions; univalent functions; local univalence; orthogonal polynomials; Meixner-Pollaczek polynomials; typically-real functions},

language = {eng},

number = {1},

pages = {45-56},

title = {Linearly-invariant families and generalized Meixner–Pollaczek polynomials},

url = {http://eudml.org/doc/268070},

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 UMCS, Mathematica

PY - 2013

VL - 67

IS - 1

SP - 45

EP - 56

AB - The extremal functions f0(z) realizing the maxima of some functionals (e.g. max |a3|, and max arg f′(z)) within the so-called universal linearly invariant family Uα (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 Pλn (x; θ,ψ) of a real variable x as coefficients of [###] 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.

LA - eng

KW - Typically-real functions; univalent functions; local univalence; orthogonal polynomials; Meixner-Pollaczek polynomials; typically-real functions

UR - http://eudml.org/doc/268070

ER -

## References

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