The continuous symmetric Hahn polynomials found in Ramanujan's lost notebook (Q555834)

Let \[ \phi(\alpha,x):=\frac{1}{\big\{1+\big(\frac{x}{\alpha}\big)^2\big\}\big\{1+\big(\frac{x}{\alpha+1}\big)^2\big\} \big\{1+\big(\frac{x}{\alpha+2}\big)^2\big\}\ldots}\,. \] For \(\alpha>0\), \(\beta>0\) and \(s\) a complex number such that \(\Re s\neq 0\) the authors give an explicit formula for the integral \[ \int_{0}^{\infty}\phi(\alpha,x)\,\phi(\beta,x)\,\frac{dx}{1+4s^2x^2}, \] in terms of gamma functions and a continued fraction. They provide two different proofs of this formula. One of them uses some results on the Hamburger moment problem.