  In previous work of the first and third authors, we proposed a conjecture
that the Kauffman bracket skein module of any knot in $S^3$ carries a natural
action of the rank 1 double affine Hecke algebra $SH_{q,t_1, t_2}$ depending on
3 parameters $q, t_1, t_2$. As a consequence, for a knot $K$ satisfying this
conjecture, we defined a three-variable polynomial invariant $J^K_n(q,t_1,t_2)$
generalizing the classical colored Jones polynomials $J^K_n(q)$. In this paper,
we give explicit formulas and provide a quantum group interpretation for the
generalized Jones polynomials $J^K_n(q,t_1,t_2)$. Our formulas generalize the
so-called cyclotomic expansion of the classical Jones polynomials constructed
by K.\ Habiro: as in the classical case, they imply the integrality of
$J^K_n(q,t_1,t_2)$ and, in fact, make sense for an arbitrary knot $K$
independent of whether or not it satisfies our earlier conjecture. When one of
the Hecke deformation parameters is set to be 1, we show that the coefficients
of the (generalized) cyclotomic expansion of $J^K_n(q,t_1)$ are determined by
Macdonald orthogonal polynomials of type $A_1$.
