  For the M\"obius spheres $S^{q,p}$, we give alternative elementary proofs of
the recursive formulas for GJMS-operators and $Q$-curvatures due to the first
author [Geom. Funct. Anal. 23, (2013), 1278-1370; arXiv:1108.0273]. These
proofs make essential use of the theory of hypergeometric series.
