An extension of typically-real functions and associated orthogonal polynomials

Abstract

Abstract. 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.

An Extension of the Chebyshev Polynomials

“…Next in [23] some properties of the generalized Chebyshev polynomials U n ( p, q; e iθ ) of the second kind was studied. These polynomials were defined by the generating function.…”

“…In [23] it was also proposed to study some generalization of the Chebyshev polynomials of the first kind, namely. 3 The Main Properties of Polynomials T n ( p, q; e iθ ) and U n ( p, q; e iθ )…”

Constrained coefficients problem for generalized typically real functions

“…We therefore, first introduce such class of functions. Definition 1.1: [5,6] Let −1 ≤ p, q ≤ 1. By T p,q we denote the class of generalized typically real functions, that is a collection of functions f ∈ H with an integral representation…”

“…The class T p,q was defined and studied in [5,6]. We underline the close relation of T p,q with the generalized Chebyshev polynomials of the second kind U n p, q; e iθ via the generating function where z ∈ D, θ ∈ 0, 2π .…”

“…We underline the close relation of T p,q with the generalized Chebyshev polynomials of the second kind U n p, q; e iθ via the generating function where z ∈ D, θ ∈ 0, 2π . [6] Some properties of the polynomials, defined by (1.6), were also studied in [7,8]. Applying (1.6) and a three-term recurrence relation it was proved that [8] U 0 p, q; e iθ = 1, U 1 p, q; e iθ = pe iθ + qe −iθ , U n+2 p, q; e iθ − pe iθ + qe −iθ U n+1 p, q; e iθ + pqU n p, q; e iθ = 0, (1.7)…”