  We investigate the homogeneous symmetric Macdonald polynomials
$P_\lambda(\X;q,t)$ for the specialization $t=q^k$. We show an identity relying
the polynomials $P_\lambda(\X;q,q^k)$ and
$P_\lambda(\frac{1-q}{1-q^k}\X;q,q^k)$. As a consequence, we describe an
operator whose eigenvalues characterize the polynomials $P_\lambda(\X;q,q^k)$.
