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

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-variable generalizations of Ramanujan’s reciprocity formula. Then we utilize also Milne’s identity to deduce a multi-variable generalization of the Askey-Wilson integral.