Generalized Lucas Theorem

Let $p$ be a prime. Let $A$ and $B$, $A\ge B\ge 0$, be integers with base $p$ expansions $A={\alpha }_i{\alpha }_{i-1}\dots {\alpha }_0$ and $B={\beta }_i{\beta }_{i-1}\dots {\beta }_0$. Lucas proved that $$\left(\genfrac{}{}{0px}{}{A}{B}\right)\equiv \prod _{j=0}^{j=i}\left(\genfrac{}{}{0px}{}{{\alpha }_j}{{\beta }_j}\right)\text{ mod }p\mathrm{.}$$ Similarly as proved by Kummer, the $p$-adic valuation $v_p\left(\genfrac{}{}{0px}{}{A}{B}\right)$ is the number of borrows when computing $A-B$ in base $p$, or the number of carries in $(A-B)+B$ in base $p$. Davis and Webb discovered a generalization of Lucas’s Theorem for prime powers. We prove a similar generalization in a different form using the concept of pseudo-digits.