Summation formulae for the bilateral basic hypergeometric series ${}_1ψ_1 ( a; b; q, z )$

We give summation formulae for the bilateral basic hypergeometric series ${}_1ψ_1( a; b; q, z )$ 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.

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