On some questions of Razpet regarding binomial coefficients (Q1861254)

Let \(A(S,p)\) denote the square matrix of order \(p^s\), \(p\) prime, where, for \(0\leq i\), \(j< p^s\), \(a_{ij}\) is the least nonnegative residue of \(\left(\begin{smallmatrix} i\\ j\end{smallmatrix}\right) \text{mod }p\) if \(i\geq j\), and \(a_{ij}= 0\) if \(i< j\). It is shown that the minimal polynomial of \(A(S,p)\) is \((s-1)^p\), thus answering a question posed by \textit{M. Razpet} [Discrete Math. 135, 377-379 (1994; Zbl 0819.05004)]. Say that an infinite matrix \(C= (c_{mn})\), \(m\geq 0\), \(n\geq 0\), has the Lucas property if, for all primes \(p\), \[ c_{mn}= \prod^k_{i=0} c_{m_in_i}(\text{mod }p), \] \[ \text{where}\quad m= \sum^k_{i=0} m_i p^i,\quad n= \sum^k_{i=0} n_ip^i,\quad 0\leq m_i,\;n_i< p. \] (Thus the matrix of binomial coefficients has the Lucas property.) Razpet asked for other matrices having the Lucas property; it is now shown that there exist uncountably many such matrices.