In this paper we give some interesting equation of $p$-adic $q$-integrals on $\mathbb Z_p$. From those $p$-adic $q$-integrals, we present a systemic study of some families of extended Carlitz $q$-Bernoulli numbers and polynomials with weight $\alpha$ in $p$-adic number field.
"A NOTE ON THE GENERALIZED HIGHER-ORDER $q$-BERNOULLI NUMBERS AND POLYNOMIALS WITH WEIGHT $\alpha$." Taiwanese J. Math. 17 (3) 785 - 800, 2013. https://doi.org/10.11650/tjm.17.2013.2396