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Tytuł artykułu

Linearly-invariant families and generalized Meixner–Pollaczek polynomials

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EN
Abstrakty
EN
The extremal functions  \(f_0(z)\)  realizing the maxima of some functionals (e.g. \(\max|a_3|\), and  \(\max{arg f^{'}(z)}\)) within the so-called universal linearly invariant family \(U_\alpha\) (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 \(P_n^\lambda(x;\theta,\psi)\) 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.
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EN
 
Rocznik
Tom
67
Numer
1
Opis fizyczny
Daty
wydano
2013
online
2015-07-15
Twórcy
Bibliografia
  • Araaya, T. K., The symmetric Meixner–Pollaczek polynomials, Uppsala Dissertations in Mathematics, Department of Mathematics, Uppsala University, 2003.
  • Chihara, T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
  • Duren, P. L., Univalent Functions, Springer, New York, 1983.
  • Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G., Higher Transcendental Functions, vol. I, McGraw-Hill Book Company, New York, 1953.
  • Golusin, G., Geometric Theory of Functions of a Complex Variable, Translations of Mathematical Monographs, no. 26, Amer. Math. Soc., Providence, R.I., 1969.
  • Ismail, M., On sieved ultraspherical polynomials I: Symmetric Pollaczek analogues, SIAM J. Math. Anal. 16 (1985), 1093–1113.
  • Kiepiela, K., Naraniecka, I., Szynal, J., The Gegenbauer polynomials and typically real functions, J. Comp. Appl. Math 153 (2003), 273–282.
  • Koekoek, R., Swarttouw, R. F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Delft University of Technology, 1998.
  • Koornwinder, T. H., Meixner–Pollaczek polynomials and the Heisenberg algebra, J. Math. Phys. 30 (4) (1989), 767–769.
  • Pommerenke, Ch., Linear-invariant Familien analytischer Funktionen, Mat. Ann. 155 (1964), 108–154.
  • Poularikas, A. D., The Mellin Transform, The Handbook of Formulas and Tables for Signal Processing, CRC Press LLC, Boca Raton, 1999.
  • Robertson, M. S., On the coefficients of typically-real functions, Bull. Amer. Math. Soc. 41 (1935), 565–572.
  • Rogosinski, W. W., Uber positive harmonische Entwicklungen und typisch-reelle Potenzreihen, Math. Z. 35 (1932), 93–121.
  • Starkov, V. V., The estimates of coefficients in locally-univalent family \(U_{\alpha}^{'}\), Vestnik Lenin. Gosud. Univ. 13 (1984), 48–54 (Russian).
  • Starkov, V. V., Linear-invariant families of functions, Dissertation, Ekatirenburg, 1989, 1–287 (Russian).
  • Szynal, J., An extension of typically-real functions, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 48 (1994), 193–201.
  • Szynal, J., Waniurski, J., Some problems for linearly invariant families, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 30 (1976), 91–102.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.ojs-doi-10_17951_a_2013_67_1_45-56
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