The Reynolds operator originated from the famous study of Reynolds in fluid mechanics in the late 19th century. The classical example of Reynolds operator is given by a special integral operator, first studied by Reynolds and Rota. Continuing a previous algebraic study of Volterra integral equations, we further reveal rich algebraic structures arising from Volterra integral equations, including following questions.

1. We show that a Volterra integral operator with separable kernel satisfies a generalized Reynolds identity, called the D-differential Reynolds identity.

2. We relate the algebraic completion from a filtration to the analytic completion from the Reynolds identity.

3. We construct free commutative complete D-differential Reynolds algebras by modifying the shuffle product.

This is a joint work with Li Guo and Richard Gustavson.