  We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured
by Haglund et al. and Bergeron et al., and proven by Carlsson and Mellit, and
Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the
combinatorial side are replaced by lattice paths lying under a line segment
whose $x$ and $y$ intercepts need not be integers, and the algebraic side is
given either by a Schiffmann algebra operator formula or an equivalent explicit
raising operator formula.
  We derive our combinatorial identity as the polynomial truncation of an
identity of infinite series of $GL_{l}$ characters, expressed in terms of
infinite series versions of LLT polynomials. The series identity in question
follows from a Cauchy identity for non-symmetric Hall-Littlewood polynomials.
