Lucas' theorem for prime powers (Q916686)

Lucas' theorem on binomial coefficients states that \(\left( \begin{matrix} A\\ B\end{matrix} \right)\equiv \left( \begin{matrix} a_ r\\ b_ r\end{matrix} \right)...\left( \begin{matrix} a_ 1\\ b_ 1\end{matrix} \right)\left( \begin{matrix} a_ 0\\ b_ 0\end{matrix} \right)(mod p)\), where p is a prime and \(A=a_ rp^ r+...+a_ 1p+a_ 0\), \(B=b_ rp^ r+...+b_ 1p+b_ 0\) are the p-adic expansions of A and B. The authors show that a similar formula holds modulo \(p^ s\) with \(s\geq 2\) where the product involves a slightly modified binomial coefficient evaluated on blocks of s digits.