Extensions of Ramanujan's reciprocity theorem and the Andrews–Askey integral

Ramanujan's reciprocity theorem may be considered as a three-variable extension of Jacobi's triple product identity. Using the method of q-partial differential equations, we extend Ramanujan's reciprocity theorem to a seven-variable reciprocity formula. Using the same method, the Andrews–Askey integral formula is extended to a q-integral formula which has seven parameters with base q.