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.
@article{bwmeta1.element.doi-10_2478_v10062-012-0021-1,
author = {Iwona Naraniecka and Jan Szynal and Anna Tatarczak},
title = {Linearly-invariant families and generalized Meixner--Pollaczek polynomials},
journal = {Annales UMCS, Mathematica},
volume = {67},
year = {2013},
pages = {45-56},
zbl = {06277079},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0021-1}
}
Iwona Naraniecka; Jan Szynal; Anna Tatarczak. Linearly-invariant families and generalized Meixner–Pollaczek polynomials. Annales UMCS, Mathematica, Tome 67 (2013) pp. 45-56. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0021-1/
[1] Araaya, T. K., The symmetric Meixner-Pollaczek polynomials, Uppsala Dissertations in Mathematics, Department of Mathematics, Uppsala University, 2003. | Zbl 1075.33001
[2] Chihara, T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. | Zbl 0389.33008
[3] Duren, P. L., Univalent Functions, Springer, New York, 1983.
[4] Erd´elyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G., Higher TranscendentalFunctions, vol. I, McGraw-Hill Book Company, New York, 1953.
[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] Ismail, M., On sieved ultraspherical polynomials I: Symmetric Pollaczek analogues, SIAM J. Math. Anal. 16 (1985), 1093-1113. | Zbl 0576.33005
[7] Kiepiela, K., Naraniecka, I., Szynal, J., The Gegenbauer polynomials and typicallyreal functions, J. Comp. Appl. Math 153 (2003), 273-282. | Zbl 1019.30013
[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] Koornwinder, T. H., Meixner-Pollaczek polynomials and the Heisenberg algebra, J. Math. Phys. 30 (4) (1989), 767-769.[Crossref] | Zbl 0672.33011
[10] Pommerenke, Ch., Linear-invariant Familien analytischer Funktionen, Mat. Ann. 155 (1964), 108-154. | Zbl 0128.30105
[11] Poularikas, A. D., The Mellin Transform, The Handbook of Formulas and Tables forSignal Processing, CRC Press LLC, Boca Raton, 1999. | Zbl 0909.94001
[12] Robertson, M. S., On the coefficients of typically-real functions, Bull. Amer. Math. Soc. 41 (1935), 565-572. | Zbl 0012.21201
[13] Rogosinski, W. W., ¨ Uber positive harmonische Entwicklungen und typisch-reellePotenzreihen, Math. Z. 35 (1932), 93-121.[Crossref] | Zbl 0003.39303
[14] Starkov, V. V., The estimates of coefficients in locally-univalent family U′α, Vestnik Lenin. Gosud. Univ. 13 (1984), 48-54 (Russian).
[15] Starkov, V. V., Linear-invariant families of functions, Dissertation, Ekatirenburg, 1989, 1-287 (Russian).
[16] Szynal, J., An extension of typically-real functions, Ann. Univ. Mariae Curie- Skłodowska, Sect. A 48 (1994), 193-201.
[17] Szynal, J., Waniurski, J., Some problems for linearly invariant families, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 30 (1976), 91-102. | Zbl 0412.30011