  Quantum K-theory of a smooth projective variety at genus zero is a collection
of integers that can be assembled into a generating series $J(Q,q,t)$ that
satisfies a system of linear differential equations with respect to $t$ and
$q$-difference equations with respect to $Q$. With some mild assumptions on the
variety, it is known that the full theory can be reconstructed from its small
$J$-function $J(Q,q,0)$ which, in the case of Fano manifolds, is a
vector-valued $q$-hypergeometric function. On the other hand, for the quintic
3-fold we formulate an explicit conjecture for the small $J$-function and its
small linear $q$-difference equation expressed linearly in terms of the
Gopakumar-Vafa invariants. Unlike the case of quantum knot invariants, and the
case of Fano manifolds, the coefficients of the small linear $q$-difference
equations are not Laurent polynomials, but rather analytic functions in two
variables determined linearly by the Gopakumar-Vafa invariants of the quintic.
Our conjecture for the small $J$-function agrees with a proposal of
Jockers-Mayr.
