Summation formulae for the bilateral basic hypergeometric series ${}_1{\psi }_1(a;b;q,z)$

We give summation formulae for the bilateral basic hypergeometric series ${}_1{\psi }_1(a;b;q,z)$ through Ramanujan’s summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian mirror symmetry on $\mathbf{R}{P}^2\times S^1$. We also show the $q\to 1-0$ limit of our summation formulae.