The paper by Shijun Liao "Notes on the homotopy analysis method: Some definitions and theorems" *Communications in Nonlinear Science and Numerical Simulation*, 14(4) 2009 pp. 983-997 may be what you're after.

The author notes in the first section of the paper:

> At the current stage of the HAM, it is urgently necessary to
> redescribe this method in a more rigorous way. So, in this paper,
> definitions of some new concepts such as the homotopy-derivative, the
> convergence-control parameter, the convergence-control vector, and so
> on, are given so as to redescribe the method more rigorously.