  We study some kinds of generalizations of Schr\"oder paths below a line with
rational slope and derive the $q$-difference equations that are satisfied by
their generating functions. As a result, we establish a relation between the
generating function of generalized Schr\"oder paths with backwards and the wave
function corresponding to colored HOMFLY-PT polynomials of torus knot
$T_{1,f}$. We also give a combinatorial proof of a recent result by
Sto\v{s}i\'c and Su{\l}kowski, in which the standard generalized Schr\"oder
paths are related to the superpolynomial of reduced colored HOMFLY-PT homology
of $T_{1,f}$.
