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## Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2013 | 67 | 1 |
Tytuł artykułu

### Linearly-invariant families and generalized Meixner–Pollaczek polynomials

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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 ) have such a form that $$f_0^{'}(z)$$  looks similar to generating function for Meixner-Pollaczek (MP) polynomials , . 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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2013
online
2015-07-15
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Bibliografia
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• 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.
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• Szynal, J., Waniurski, J., Some problems for linearly invariant families, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 30 (1976), 91–102.
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