In Ramanujan’s "Lost Notebook" there are a few definite integrals along with their values. Two of these claims were proved by the reviewer, who closed his paper by saying he did not know what the real interest is in these integrals. However he suspected the results could be understood as well as proved. In the present paper, the author shows that both integrals are q-extensions of beta integrals.
In one case the integral can be moved from the real line to the unit circle and the author finds two sets of polynomials which are biorthogonal on the unit circle with respect to this measure. When the weight function is even the biorthogonal polynomials become the orthogonal polynomials on the unit circle introduced in the comment to one of G. Szegö’s papers, in his Collected Papers (1982; Zbl 0491.01016) (paper 26-6). In the case the biorthogonal polynomials were also introduced in a comment to another of Szegö’s papers in the same book. The general q-case is new.
For the other integral, the integrand is the weight function for a finite set of biorthogonal Laurent polynomials, and in one case these polynomials exist for all degrees. As far as I know, this is the first instance of explicit biorthogonal Laurent polynomials whose weight function is positive.