Macdonald polynomials at t=qk

We investigate the t=qk specialization of the homogeneous symmetric Macdonald polynomials Pλ(X;q,t). We show an identity relating the polynomials Pλ(X;q,qk) and . As a consequence, we describe an operator whose eigenvalues characterize the polynomials Pλ(X;q,qk).