Summary: We first extend the digital binomial identity as given by Nguyen et al. to an identity in an arbitrary base $b$, by introducing the $b$-ary binomial coefficients. Then, we study the properties of these coefficients such as their orthogonality, their link with Lucas theorem and their extension to multinomial coefficients. Finally, we analyze the structure of the corresponding $b$-ary Pascal-like triangles.

05C30, 05C78

b-ary expansion, b-ary binomial coefficient, Lucas theorem