  For an integer $q\ge2$, a $q$-recursive sequence is defined by recurrence
relations on subsequences of indices modulo some powers of~$q$. In this
article, $q$-recursive sequences are studied and the asymptotic behavior of
their summatory functions is analyzed. It is shown that every $q$-recursive
sequence is $q$-regular in the sense of Allouche and Shallit and that a
$q$-linear representation of the sequence can be computed easily by using the
coefficients from the recurrence relations. Detailed asymptotic results for
$q$-recursive sequences are then obtained based on a general result on the
asymptotic analysis of $q$-regular sequences.
  Three particular sequences are studied in detail: We discuss the asymptotic
behavior of the summatory functions of Stern's diatomic sequence, the number of
non-zero elements in some generalized Pascal's triangle and the number of
unbordered factors in the Thue--Morse sequence. For the first two sequences,
our analysis even leads to precise formul\ae{} without error terms.
