Some properties of a hypergeometric function which appear in an approximation problem

Abstract

In this paper we consider properties and power expressions of the functions $f:(-1,1)\rightarrow \mathbb{R }$ and $f_L:(-1,1)\rightarrow \mathbb{R }$, defined by

$$\begin{aligned} f(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma }{\sqrt{1-t^2}}\,\mathrm{d}t \quad \text{ and}\quad f_L(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma \log (1+x t)}{\sqrt{1-t^2}}\,\mathrm{d}t, \end{aligned}$$

respectively, where $\gamma$ is a real parameter, as well as some properties of a two parametric real-valued function $D(\,\cdot \,;\alpha ,\beta ) :(-1,1) \rightarrow \mathbb{R }$, defined by

$$\begin{aligned} D(x;\alpha ,\beta )= f(x;\beta )f(x;-\alpha -1)- f(x;-\alpha )f(x;\beta -1),\quad \alpha ,\beta \in \mathbb{R }. \end{aligned}$$

The inequality of Turán type

$$\begin{aligned} D(x;\alpha ,\beta )>0,\quad -1<x<1, \end{aligned}$$

for $\alpha +\beta >0$ is proved, as well as an opposite inequality if $\alpha +\beta <0$. Finally, for the partial derivatives of $D(x;\alpha ,\beta )$ with respect to $\alpha$ or $\beta$, respectively $A(x;\alpha ,\beta )$ and $B(x;\alpha ,\beta )$, for which $A(x;\alpha ,\beta )=B(x;-\beta ,-\alpha )$, some results are obtained. We mention also that some results of this paper have been successfully applied in various problems in the theory of polynomial approximation and some “truncated” quadrature formulas of Gaussian type with an exponential weight on the real semiaxis, especially in a computation of Mhaskar–Rahmanov–Saff numbers.