I will introduce recursion relations for partition functions with values in the HOMFLYPT skein of a solid torus and explain how such recursion relations naturally arise in the context of the skein-valued Gromov-Witten invariants of Ekholm and Shende. As an application, I give a proof of a skein-valued lift of the pentagon relation for q-Pochhammers and derive a factorization of the colored HOMFLYPT invariants of the Hopf link reminiscent of the Gopakumar-Vafa formula. Part of this talk is based on joint work with T. Ekholm and P. Longhi.