Extensions of Bailey's $_6ψ_6$ series identity with seven free parameters

Chuanan Wei

Published 2019-05-06Version 1

In the literature of basic hypergeometric series, Bailey's $_6\psi_6$ series identity is very important. So finding the nontrivial extension of it is a quite significative work. In this paper, we establish, above all, a transformation formula involving two $_8\psi_8$ series and a $_8\phi_7$ series according to the analytic continuation argument. When the parameters are specified, it reduces to Bailey's $_6\psi_6$ series identity and gives two different bilateral generalizations of the nonterminating form of Jackon's $_8\phi_7$ summation formula. Subsequently, a new extension of Ramanujan's $_1\psi_1$ series identity involving two $_2\psi_2$ series and a $_2\phi_1$ series is derived from this formula and a known transformation formula for bilateral basic hypergeometric series. Finally, a new proof for the known extension of Bailey's $_6\psi_6$ series identity involving a $_8\psi_8$ series and two $_8\phi_7$ series is also offered via the analytic continuation argument.