A special class of Jacobi series and some applications

AbstractIn this paper, series of the form F(cosθ) ∼ ∑n=2∞ωn(α,β)nγ(logn)δrn(α,β)(cosθ) are considered, where Rn(α,β)(x) denotes the Jacobi polynomial normalized to be 1 at x = 1 and ωn(α,β) = (2n + α + β + 1) Γ(n + α + β + 1) Γ(n + α + 1)Γ(n + β + 1) Γ(n + 1) Γ(α + 1) Γ(α + 1) = 0(n2α + 1). It turns out, that for 0 < θ ⩽ π, the function F(cos θ) is continuous. As θ → 0+, its behavior is given by F(cos θ) ∼- Γ(α +1 −, γ2)/Γ(γ2) Γ(α + 1) (sin, θ2)γ − 2α − 2 (log θ−1)−δ, if 0 < γ < 2α + 2. If γ = 0 the results are slightly different. Next, fractional integration and differentiation for Jacobi series are introduced and the above results are used to show that the classical theorems on fractional integration and differentiation can be carried over. As an application sufficient conditions are given for F(cos θ) to have a uniformly convergent or an absolutely convergent Fourier-Jacobi series. This paper will be followed by another one in which the results of this paper are used to deal with approximation theorems for Jacobi series