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We give new proofs for certain bilateral basic hypergeometric summationformulas using the symmetries of the corresponding series. In particular, wepresent a proof for Bailey's $_3\psi_3$ summation formula as an application. Wealso prove a multiple series analogue of this identity by consideringhyperoctahedral group symmetries of higher ranks.

Multilateral basic hypergeometric summation identities and hyperoctahedral group symmetries