On the arithmetic complexity of some linear mappings

We investigate the arithmetic complexity of computation of some linear maps originated from combinatorics and number theory (e.g. binomial map, $q$-bimomial Gauss's map, Stirling's maps of both kinds, Smith's map with the matrix made up from GCD's of indices of rows and columns). The arithmetic complexity here is the minimal required number of additive operations or, alternatively, additive operations and multiplications, or, generally, all arithmetic operations including division. The computation starts from the constant $1$.