MIN-turns and MAX-turns in k-Dyck paths: a pure generating function approach

$k$-Dyck paths differ from ordinary Dyck paths by using an up-step of length $k$. We analyze at which level the path is after the $s$-th up-step and before the $(s+1)$st up-step. In honour of Rainer Kemp who studied a related concept 40 years ago the terms $\text{textsc}max$-terms and $\text{textsc}min$-terms are used. Results are obtained by an appropriate use of trivariate generating functions; practically no combinatorial arguments are used.