Double basic hypergeometric sums via a regularized Jackson $q$-integral

Using the Andrews–Askey integral, we derive a holomorphic extension formula for a Jackson $q$-integral. This formula allows identities established for terminating specializations to be continued to the nonterminating case. Our first application yields two companion binomial-type double-sum formulas. The second formula contains, as special cases, the nonterminating Sears–Carlitz transformation of Gasper and Rahman and Rahman’s generating function for the Askey–Wilson polynomials. Applying the same method to the $q$-Chu–Vandermonde and Rogers’ ${}_6{\phi }_5$ summations, we obtain two further integral representations for double sums of basic hypergeometric type. A balanced ${}_4{\phi }_3$ specialization of the latter identity is equivalent, via Sears’ transformation, to the double-series transformation of Ismail, Rahman, and Suslov.