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.
Edit: see M.Turkyilmazoglu, "A note on the homotopy analysis method" Applied Math. Letters, 23 (2010) pp. 1226-1230 for an extension of some of the theorems in Liao's paper.