## Michael J. Schlosser

# Elliptic enumeration of nonintersecting lattice paths

### (17 pages)

**Abstract.**
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 _{10}V_{9} 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 _{8}Φ_{7} and Dougall's
_{7}F_{6} summation.
By considering nonintersecting lattice paths we are led to
a multivariate extension of the _{10}V_{9} summation
which turns out to be a special case of an identity originally
conjectured by
Warnaar,
later proved by
Rosengren.
We conclude with discussing some future perspectives.

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