Abstract
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| 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.
Highlights
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
The order of the linearly-invariant family M is defined as ord M = sup |a2(f )|
It is well known that α ≥ 1 and U1 ≡ Sc = the class of convex univalent functions in D, and the familiar class S of all univalent functions is strictly included in U2
Summary
We will find the threeterm recurrence relation, the explicite formula, the hypergeometric representation and the difference equation for (GMP) polynomials Pnλ(x; θ, ψ). (i) The (MP) polynomials Pnλ(x; θ) satisfy the three-term recurrence relation: P−λ1(x; θ) = 0, P0λ(x; θ) = 1, nPnλ(x; θ) = 2[xsinθ + (n − 1 + λ) cos θ]Pnλ−1(x; θ). −n, λ + ix, 2λ; 1 − e−2iθ (iiii) The polynomials y(x) = Pnλ(x; θ) satisfy the following difference equation eiθ(λ − ix)y(x + i) + 2i[x cos θ − (n + λ)sinθ]y(x) − e−iθ(λ + ix)y(x − i) = 0. −n, λ + ix, 2λ; 1 + e−2iθ (iiii) The polynomials y(x) = Qλn(x; θ) satisfy the following difference equation eiθ(λ − ix)y(x + i) − 2[xsinθ + (n + λ) cos θ]y(x) + e−iθ(λ + ix)y(x − i) = 0.
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