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. 
Hironori Mori and Takeshi Morita
math.CA hep-th