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 T1,f. We also give a combinatorial proof of a recent result by Sto{s}i\'c and Su{}kowski, in which the standard generalized Schr\"oder paths are related to the superpolynomial of reduced colored HOMFLY-PT homology of T1,f.
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Ji et al. (2024) studied this question.
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