An analog of the Dougall formula and of the de Branges--Wilson integral

We derive a beta-integral over $\mathbb{Z}\times \mathbb{R}$ , which is a counterpart of the Dougall ${}_5{H}_5$-formula and of the de Branges–Wilson integral, our integral includes ${}_{10}{H}_{10}$-summation. For a derivation we use a two-dimensional integral transform related to representations of the Lorentz group, this transform is a counterpart of the Olevskii index transform (a synonym: Jacobi transform).