General Ramanujan-type integral representations for bilateral basic hypergeometric series

Determine whether more general Ramanujan-type integral representations exist for bilateral basic hypergeometric series (such as the bilateral basic hypergeometric series {}_rψ_s), meaning real-line integral formulas with integrands built from products of q-Pochhammer symbols (or q-gamma-type factors) analogous to the classical Ramanujan-type beta integrals, that persist in the limit q→1⁻ and recover the corresponding integral representations for bilateral hypergeometric series.

Background

The work develops integral representations for classical bilateral hypergeometric series via the Poisson summation formula and studies when these integrals reduce to beta integrals. In Section 6, q-extensions are explored: a q-analogue of Ramanujan’s Fourier transform is established, integral representations for {}_mψ_m are derived, and a q-beta integral linked to a very-well-poised {}_6ψ_6 is presented, together with several q→1⁻ limits that recover classical results in specific cases.

However, not all degenerations yield nontrivial limits, and the authors note that the basic (q-)case is not fully parallel to the classical case. Motivated by these partial correspondences, they pose an explicit open question about the existence of more general Ramanujan-type integral representations for basic bilateral series that mirror the integral representations available in the q→1⁻ limit for bilateral hypergeometric series.