Sums of products of q-Bernoulli numbers

Abstract. 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 $ \sum\limits_{r=i_1 +\cdots +i_1} \sum\limits_{k_1=0}^{r-i_1} \sum\limits_{k_2=0}^{r-i_1-i_2} \cdots \sum\limits_{k_{l-1}=0}^{r-i_1-i_2-\cdots -i_{l-1}} {r \choose {i_1,\cdots,i_l}} {{r-i_1} \choose {k_1}} \cdots {{r-i_1-\cdots -i_{l-1}} \choose {kl_1}}\times \beta_{{k_1}+{i_1}} (\alpha_1 ,q)\beta_{{k_2}+{i_2}} (\alpha_2 ,q)\cdots \beta_{{k_l-1}+{i_l-1}} (\alpha_{l-1},q)\beta_{i_l} (\alpha_1 ,q)(q-1)^{k_1 +\cdots +k_{l-1}} $ where $ \beta _m(\alpha ,q) $ is the Carlitz's q-Bernoulli polynomial.