Properties of the Dzyadyk polynomial kernels on a closed segment (Q807882)

The author proves some properties of the following polynomial kernel \[ {\mathcal D}_{m,n,r}(y,x):=\frac{1}{(m-1)!}\frac{\partial^ m}{\partial x^ m}[(x-y)^{m-1}\int^{\beta +\alpha}_{\beta - \alpha}J_{n,r}(t)dt], \] where \(\alpha:=\arccos y\), \(\beta:=\arccos x\) and \[ J_{n,r}(t):=\frac{1}{\gamma_{n,r}}(\frac{\sin (nt/2)}{\sin (t/2)})^{2r+2},\quad \gamma_{n,r}:=\int^{\pi}_{-\pi}(\frac{\sin (nt/2)}{\sin (t/2)})^{2r+2} dt. \]