Subblock Occurrences in the q-Ary Representation of n

Let $B_q ( w,n )$ denote the number of subblocks w in the q-ary representation of $n \in \mathbb{N}$ (overlapping allowed). The paper deals with the mean value $m^{ - 1} \cdot \sum_{n = 0}^{m - 1} B_q ( w,n ) $ and an application of this result on the summing function of the generalized “sum of digits” function introduced by H. Prodinger [SIAM J. Alg. Discr. Meth., 3 (1982), pp. 35–42].