Sums of products of q-Bernoulli numbers

We define Carlitz's q-Bernoulli number of higher order using an integral by the q-analogue mu (q) [4] of the ordinary p-adic invariant measure. Using q-Bernoulli number of higher order, we give the formula for sums of products of Carlitz's q-Bernoulli numbers of the form [GRAPHICS] x beta (k1+i1)(alpha (1),q)beta (k2+i2) (alpha (2), q) . . . beta (kl-1+l-1) (alpha (l-1), q)beta (il)(alpha (l),q)(q - 1)(k1 + . . . + kl-1), where beta (m)(alpha, q) is the Carlitz's q-Bernoulli polynomial.