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Ramanujan's reciprocity theorem may be considered as a three-variableextension of Jacobi's triple product identity. Using the method of $q$-partialdifferential equations, we extend Ramanujan's reciprocity theorem to aseven-variable reciprocity formula. The Andrews--Askey integral is a$q$-integral having four parameters with base $q$. Using the same method weextend the Andrews--Askey integral formula to a $q$-integral formula which hasseven parameters with base $q$.

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