Let $E$ be an elliptic curve defined over a finite field $\mathbf{F}_{q}$, $q = p^{d}$, $p > 3$, and a prime number $\ell > 3$ such that $q \equiv 1 \pmod{\ell}$ and $\ell \mid \# E(\mathbf{F}_{q})$. In this paper we study the possible factorisation patterns over $\mathbf{F}_{q}[x]$ of the $\ell^{k}$-division polynomials associated to $E$ with $k \geq 2$, extending the work of Verdure [6] for $k=1$.

In 1977, Gosper introduced a conjectural evaluation for a hypergeometric series that has been described as strange by a number of authors. In 2013, Ebisu proved Gosper’s conjecture using contiguity operators. Subsequently, in 2017, Chu provided another proof of Gosper’s conjecture, using a telescoping argument together with Pfaff’s transformation. In this article, we present a new and simplified proof of Gosper’s conjecture that is inequivalent to the previous proofs due to Ebisu and Chu. Our proof relies on an evaluation technique that was previously given by Campbell and Cantarini and that involves the modified Abel lemma on summation by parts. We also show how this method may be applied to prove generalizations and variants of Gosper’s summation.