Linearly-invariant families and generalized Meixner–Pollaczek polynomials

Abstract

Abstract. 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 λ n (x; θ, ψ) of a real variable x as coefficients ofwhere 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 MeixnerPollaczek 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 λ (x; θ, ψ; z) is a kernel, will be discussed. Linearly-invariant families of holomorphic functions(1.1) f (z) = z + a 2 z 2 + . . . , z ∈ D 2010 Mathematics Subject Classification. 30C45, 30C70, 42C05, 33C45.