On the Deuring Polynomial for Drinfeld Modules in Legendre Form

We study a family ${\psi }^{\lambda }$ of ${\mathbb{F}}_q[T]$-Drinfeld modules, which is a natural analog of Legendre elliptic curves. We then find a surprising recurrence giving the corresponding Deuring polynomial $H_{p(T)}(\lambda )$ characterising supersingular Legendre Drinfeld modules ${\psi }^{\lambda }$ in characteristic $p(T)$.