Michael J. Schlosser
Elliptic enumeration of nonintersecting lattice paths
We enumerate lattice paths in the planar integer lattice consisting of
positively directed unit vertical and horizontal steps with respect to
a specific elliptic weight function.
The elliptic generating function of paths from a given starting point
to a given end point evaluates to an elliptic generalization of the
binomial coefficient. Convolution gives an identity equivalent to
Frenkel and Turaev's 10V9 summation.
This appears to be the first combinatorial proof of the latter,
and at the same time of some important degenerate cases including
Jackson's 8φ7 and Dougall's
By considering nonintersecting lattice paths we are led to
a multivariate extension of the 10V9 summation
which turns out to be a special case of an identity originally
later proved by
We conclude with discussing some future perspectives.
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