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Linearly-invariant families and generalized Meixner–Pollaczek polynomials

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Naraniecka I., Szynal J., Tatarczak A.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2013

tom 67

nr 1

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.

An extension of typically-real functions and associated orthogonal polynomials

100%

Naraniecka I., Szynal J., Tatarczak A.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2011

tom 65

nr 2

Two-parameters extension of the family of typically-real functions is studied. The definition is obtained by the Stjeltjes integral formula. The kernel function in this definition serves as a generating function for some family of orthogonal polynomials generalizing Chebyshev polynomials of the second kind. The results of this paper concern the exact region of local univalence, bounds for the radius of univalence, the coefficient problems within the considered family as well as the basic properties of obtained orthogonal polynomials.

Ograniczanie wyników

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

2 Naraniecka I.

2 Szynal J.

2 Tatarczak A.

1 2013

1 2011

Wyniki wyszukiwania

Linearly-invariant families and generalized Meixner–Pollaczek polynomials

An extension of typically-real functions and associated orthogonal polynomials