I am unable to understand the notation of equations (1.1) and (1.6) in page 2 of Kowalski and Belger's paper "Riemannian metric with the prescribed curvature tensor and all its covariant derivatives at one point" (MSN). My intuition is that (1.6) should be iterated derivatives of the special case when $k=0$ but I still couldn't figure out what the notation of the author is supposed to mean. I would be grateful if someone can help.
From what you've written, you're obviously familiar with the idea that a linear map $a:V\to V$ has a natural action $\mu\mapsto a\cdot \mu$ on the algebra of covariant tensors given by
$$ (a\cdot \mu)(v_1,\dots,v_k) = \sum_{i=1}^k \mu(v_1,\dots,v_{i-1},av_i,v_{i+1}\dots, v_k) $$
My hunch is that notation (1.1) is designed to provide a compact way of writing `partial actions' of endomorphisms on tensors, so that rather than summing over all slots, we just sum a few. In particular, I think
$$ R^{(k-i)}_{Z_k\dots Z_{i+1}}(R_{AB}^{(0)}\cdot R^{(i)})_{Z_i\dots Z_1}(X,Y,U,V) $$
probably means:
$$ \begin{array}{llll} R^{(k)}(R_{AB}^{(0)}X,Y,U,V,Z_1,\dots, Z_k) + \\ R^{(k)}(X,R_{AB}^{(0)}Y,U,V,Z_1,\dots, Z_k) + \\ R^{(k)}(X,Y,R_{AB}^{(0)}U,V,Z_1,\dots, Z_k) + \\ R^{(k)}(X,Y,U,R_{AB}^{(0)}V,Z_1,\dots, Z_k) + \\\sum_{j=1}^i R^{(k)}(X,Y,U,V,Z_1,\dots,Z_{j-1},R_{AB}^{(0)}Z_j,Z_{j+1},\dots,Z_i,Z_{i+1},\dots,Z_k) \end{array} $$
which is like $R_{AB}^{(0)}\cdot R^{(k)}$ but the with terms involving $(R_{AB}^{0}Z_j)_{j=i+1}^k$ omitted. One way to determine whether my hunch is correct is to check whether this last expression equals the left hand side of (1.6) when $R^{(k)} = \nabla^k R$ for a genuine curvature tensor $R$ (I unfortunately don't know enough about identities involving curvature tensors to be able to do this quickly).
