The Gegenbauer polynomials and typically real functions

Kiepiela, Katarzyna; Naraniecka, Iwona; Szynal, Jan

doi:10.1016/s0377-0427(02)00642-8

Cited by 34 publications

(20 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[16], [7]) and T(1, 0) = T(λ, τ ). In parallel way we are going to study the extremal problems within the classes T(λ, τ ) and T (λ, τ ), λ > 0, τ ∈ R of holomorphic functions of the form (1.1) which have the integral representation f (z) = The classes T(λ, τ ), T(λ, τ ) and T (λ, τ ) differ pretty much, for instance all coefficients a k of f ∈ T(λ, τ ) are real, however the odd coefficients of f ∈ T(λ, τ ) are real and even coefficients of f ∈ T(λ, τ ) are purely imaginary.…”

Section: Remarkmentioning

confidence: 99%

Linearly-invariant families and generalized Meixner–Pollaczek polynomials

Naraniecka

Szynal

Tatarczak

2013

Annales UMCS, Mathematica

View full text Add to dashboard Cite

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.

show abstract

Section: Remarkmentioning

confidence: 99%

Linearly-invariant families and generalized Meixner–Pollaczek polynomials

Naraniecka

Szynal

Tatarczak

2013

Annales UMCS, Mathematica

View full text Add to dashboard Cite

show abstract

“…One of the distinctive cases of orthogonal polynomials is Gegenbauer polynomials. These polynomials are endowed with typically real functions T R due to the relation T R = coS R [14]. This relation has an impressive role in the theory of geometric functions in estimating coefficient bounds.…”

Section: Introductionmentioning

confidence: 99%

On a subclass of bi-close-to-convex functions by means of the Gegenbauer polynomial

Alamoush¹,

Bulut²

2022

Math. Appl.

View full text Add to dashboard Cite

show abstract

“…A special case of orthogonal polynomials is Gegenbauer polynomials. They are representatively related with typically real functions T R as discovered in [4], where the integral representation of typically real functions and generating function of Gegenbauer polynomials are using common algebraic expressions. Undoubtedly, this led to several useful inequalities appear from the Gegenbauer polynomial realm.…”

Section: Introductionmentioning

confidence: 99%