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

## TL;DR

This paper introduces a new beta-integral over a mixed discrete-continuous domain, extending classical hypergeometric identities using a Lorentz group-related integral transform.

## Contribution

It derives a novel beta-integral that generalizes the Dougall and de Branges--Wilson formulas through a Lorentz group-based integral transform.

## Key findings

- Includes $_{10}H_{10}$-summation within the integral
- Establishes a new connection between hypergeometric identities and Lorentz group representations
- Provides a new integral transform related to the Jacobi transform

## Abstract

We derive a beta-integral over $\mathbb{Z}\times \mathbb{R}$ , which is a counterpart of the Dougall $_5H_5$-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).