The $(2,3)$-generation of the finite simple odd-dimensional orthogonal groups

The complete classification of the finite simple groups that are $(2,3)$-generated is a problem which is still open only for orthogonal groups. Here, we construct $(2,3)$-generators for the finite odd-dimensional orthogonal groups ${\mathrm{\Omega }}_{2k+1}(q)$, $k\ge 4$. As a byproduct we also obtain $(2,3)$-generators for ${\mathrm{\Omega }}_{4k}^{+}(q)$ with $k\ge 3$ and $q$ odd, and for ${\mathrm{\Omega }}_{4k+2}^{\pm }(q)$ with $k\ge 4$ and $q\equiv \pm 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$.