Linearly-invariant families and generalized Meixner – Pollaczek polynomials

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 P n (x; θ, ψ) of a real variable x as coefficients of G(x; θ, ψ; z) = 1 (1− zeiθ)λ−ix(1− zeiψ)λ+ix = ∞ ∑ n=0 P n (x; θ, ψ)z , |z| < 1, 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 zG(x; θ, ψ; z) is a kernel, will be discussed. 1. Linearly-invariant families of holomorphic functions (1.1) f(z) = z + a2z + . . . , z ∈ D 2010 Mathematics Subject Classification. 30C45, 30C70, 42C05, 33C45.

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 λ (x; θ, ψ; z) is a kernel, will be discussed.
A family M of holomorphic functions of the form (1.1) is linearly-invariant if it satisfies two conditions: (a) f (z) = 0 for any z in D (local univalence), (b) for any linear fractional transformation φ(z) = e iθ z + a 1 + āz , a, z ∈ D, θ ∈ R, of D onto itself, the function The order of the linearly-invariant family M is defined as Universal invariant family U α is defined as It is well known that α ≥ 1 and U 1 ≡ S c = the class of convex univalent functions in D, and the familiar class S of all univalent functions is strictly included in U 2 .Moreover, for every α > 1, the class U α contains functions which are infinitely valent in D [10], for example: Another example of such a function was presented in [15]: for which Functions of the form (1.2) appear to be extremal for the long lasting problems: max recently solved by Starkov [14], [15], who proved that the extremal function for max |a 3 | is of the form (1.2) with t 1 = θ, t 2 = −θ, where However, the extremal function 15]).
We see that the extremal function for max f ∈Uα |a 3 | has a special form leading to (MP) polynomials, but the extremal function for max f ∈Uα | arg f (z)| leads to (GMP) polynomials, defined below.
(ii) The Cauchy product for the power series and
(iiii) Putting (x + i) and (x − i) instead of x into the generating function (2.1), we find that Differentiation of the generating function (2.1) with respect to z and comparison of the coefficients at z n−1 yields: which together with (2.6) gives (2.5).
The first four polynomials P λ n are given by the formulas: Corollary 1.

Corollary 3. (i)
The (QMP) polynomials Q λ n = Q λ n (x; θ) satisfy the threeterm recurrence relation: are given by the formula: (iiii) The polynomials y(x) = Q λ n (x; θ) satisfy the following difference equation The polynomials S λ n = S λ n (x; θ) are given by the formula: (iii) The polynomials S λ n = S λ n (x; θ) have the hypergeometric representation (iiii) The polynomials y(x) = S λ n (x; θ) satisfy the following difference equation Theorem 2.2.The polynomials Q λ n (x; θ) are orthogonal on (−∞, +∞) with the weight and In the proof we use the following lemmas.

Lemma 4. For arbitrary polynomial
Proof.Using hypergeometric representation for Q λ n (x; θ) we can write (4 cos 2 θ) Using the well-known formula: Therefore by (2.7) Using hypergeometric representation for Q λ n (x; θ) we can write where which ends the proof after some obvious simplifications.
Remark 1.In the case x = 0 we can obtain "more pleasant" sets of "polynomials": for which one can prove the following.
In special case τ = 0, λ = 1, i.e.T = T(1, 0), we are able to find explicitly the radius of local univalence and the radius of univalence of T which differ from the corresponding values in the class T = T(1, 0).
The same remarks concern also the sets of polynomials S 0 (x, θ) = lim where µ is a probability measure on ∆ = (0, π) × R.