A lonesum matrix is a matrix that can be uniquely reconstructed from its row
and column sums. Kaneko defined the poly-Bernoulli numbers Bm(n) by a
generating function, and Brewbaker computed the number of binary lonesum
m×n-matrices and showed that this number coincides with the
poly-Bernoulli number Bm(−n). We compute the number of q-ary lonesum
m×n-matrices, and then provide generalized Kaneko's formulas by using
the generating function for the number of q-ary lonesum m×n-matrices.
In addition, we define two types of q-ary lonesum matrices that are composed
of strong and weak lonesum matrices, and suggest further researches on lonesum
matrices. \Comment: 27 page