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By using two known transformation formulas for basic hypergeometric series,we establish a direct extension of Bailey's $_6\psi_6$-series identity.Subsequently, it and Milne's identity are employed to drive multi-variablegeneralizations of Ramanujan's reciprocity formula. Then we utilize alsoMilne's identity to deduce a multi-variable generalization of the Askey-Wilsonintegral.

Generalizations of Ramanujan's reciprocity formula and the Askey-Wilson integral