A non-standard Fourier expansion for the Drinfeld discriminant function (Q992018)

Let \(A={\mathbb F}_q[T]\), \(K={\mathbb F}_q(T)\), \(K_\infty={\mathbb F}_q\left(\left(\frac 1T\right)\right)\), \(C\) the completed algebraic closure of \(K_\infty\). The Carlitz module determines a rank one \(A\)-module in \(C\) generated by an element \(\overline{\pi}\). Let \(t(z)=\frac{1}{\overline{\pi}} \sum_{a\in A}\frac{1}{z+a}\). For the Drinfeld discriminant function \(\Delta(z)\) the author proves that \[ \overline{\pi}^{1-q^2}\Delta(z)=-\sum_{\substack{ a\in A\\ a\,\text{monic}}} a^{q(q-1)}t(az)^{q-1}. \]