  We provide analytic proofs for the shape invariance of the recently
discovered (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417) two families of
infinitely many exactly solvable one-dimensional quantum mechanical potentials.
These potentials are obtained by deforming the well-known radial oscillator
potential or the Darboux-P\"oschl-Teller potential by a degree \ell
(\ell=1,2,...) eigenpolynomial. The shape invariance conditions are attributed
to new polynomial identities of degree 3\ell involving cubic products of the
Laguerre or Jacobi polynomials. These identities are proved elementarily by
combining simple identities.
