A family of vector-valued quantum modular forms of depth two

Understand

We introduce and investigate an infinite family of functions which are shown to have generalised quantum modular properties.

• We realise their "companions" in the lower half plane both as double Eichler integrals and as non-holomorphic theta functions with coefficients given by double error functions.
• Further, we view these Eichler integrals in a modular setting as parts of certain weight two indefinite theta series.