A recurring qq-integral formula

In this paper, we use the qq-Chu–Vandermonde formula to derive a recurring qq-integral formula. Some applications of the recurring qq-integral formula are also given, which include deriving the 3ϕ23ϕ2 representation of the Al-Salam–Carlitz polynomials and the extension of the terminating Sears’ 3ϕ23ϕ2 transformation formula.