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

Abstract

top
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.

How to cite

top

Iwona 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

top
  1. [1] Araaya, T. K., The symmetric Meixner-Pollaczek polynomials, Uppsala Dissertations in Mathematics, Department of Mathematics, Uppsala University, 2003. Zbl1075.33001
  2. [2] Chihara, T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. Zbl0389.33008
  3. [3] Duren, P. L., Univalent Functions, Springer, New York, 1983. 
  4. [4] Erd´elyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G., Higher TranscendentalFunctions, vol. I, McGraw-Hill Book Company, New York, 1953. 
  5. [5] Golusin, G., Geometric Theory of Functions of a Complex Variable, Translations of Mathematical Monographs, no. 26, Amer. Math. Soc., Providence, R.I., 1969. 
  6. [6] Ismail, M., On sieved ultraspherical polynomials I: Symmetric Pollaczek analogues, SIAM J. Math. Anal. 16 (1985), 1093-1113. Zbl0576.33005
  7. [7] Kiepiela, K., Naraniecka, I., Szynal, J., The Gegenbauer polynomials and typicallyreal functions, J. Comp. Appl. Math 153 (2003), 273-282. Zbl1019.30013
  8. [8] Koekoek, R., Swarttouw, R. F., The Askey-scheme of hypergeometric orthogonalpolynomials and its q-analogue, Report 98-17, Delft University of Technology, 1998. 
  9. [9] Koornwinder, T. H., Meixner-Pollaczek polynomials and the Heisenberg algebra, J. Math. Phys. 30 (4) (1989), 767-769.[Crossref] Zbl0672.33011
  10. [10] Pommerenke, Ch., Linear-invariant Familien analytischer Funktionen, Mat. Ann. 155 (1964), 108-154. Zbl0128.30105
  11. [11] Poularikas, A. D., The Mellin Transform, The Handbook of Formulas and Tables forSignal Processing, CRC Press LLC, Boca Raton, 1999. Zbl0909.94001
  12. [12] Robertson, M. S., On the coefficients of typically-real functions, Bull. Amer. Math. Soc. 41 (1935), 565-572. Zbl0012.21201
  13. [13] Rogosinski, W. W., ¨ Uber positive harmonische Entwicklungen und typisch-reellePotenzreihen, Math. Z. 35 (1932), 93-121.[Crossref] Zbl0003.39303
  14. [14] Starkov, V. V., The estimates of coefficients in locally-univalent family U′α, Vestnik Lenin. Gosud. Univ. 13 (1984), 48-54 (Russian). 
  15. [15] Starkov, V. V., Linear-invariant families of functions, Dissertation, Ekatirenburg, 1989, 1-287 (Russian). 
  16. [16] Szynal, J., An extension of typically-real functions, Ann. Univ. Mariae Curie- Skłodowska, Sect. A 48 (1994), 193-201. 
  17. [17] Szynal, J., Waniurski, J., Some problems for linearly invariant families, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 30 (1976), 91-102. Zbl0412.30011

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.