The map behind a binomial coefficient matrix over \({\mathbb{Z}}/p{\mathbb{Z}}\) (Q1112918)

Let \(q=p^ n\), where p is a prime and let \(F_ q\) be the field with q elements. Let J be the \(q\times q\) matrix over \(F_ q\) whose (i,j) entry is the binomial coefficient \(\left( \begin{matrix} i+j\\ j\end{matrix} \right)\), \(0\leq i,j\leq q-1\). In a recent paper [ibid. 90, 65-72 (1987; Zbl 0617.15010)] \textit{N. Strauss} proved by use of generating functions, that \(J^ 3=I\). Strauss' result follows directly (Corollary 1) from the following theorem proved in the present paper: Theorem 1. Let V be the vector space of all functions from \(F_ q\) to itself. For all integers j, \(0\leq j\leq q-1\), let \(f_ j\) be the polynomial function defined by \(f_ j(x)=x^ j\) so that \(\{f_ j\}\) is a basis for V over \(F_ q\). Let T: \(V\to V\) be the linear mapping defined by \((Tf)(x)=f(1/(1-x))\) for all \(f\in V\). Then the matrix of T with respect to the basis \(\{f_ j\}\) is precisely J. The proof of this theorem requires extending elements \(f\in V\) to the ``projective line'' \(F_ q\cup \{\infty \}\) by defining \(f(\infty)=-\sum_{x\in F_ q}f(x)\) and enlarging V by defining, for \(g(x)=1/(1-x)\) all \(1\neq x\in F_ q\), \(g(1)=\infty\) and \(g(\infty)=0\). The author also determines, as Strauss did in his earlier paper, the multiplicities of the eigenvalues of J.