A Combinatorial Correspondence for Walks in Weyl Chambers

The m m determinant of hyperbolic Bessel functions det jI a i ?b j (2x)j can be factored into two smaller determinants by elementary operations if a i = ?a m+1?i and b i = ?b m+1?i. We give combinatorial interpretations for these determinants as exponential generating functions for walks which stay within Weyl chambers. We then use these to provide a combinatorial proof of the formulas by nding a sequence of reeections which give a correspondence between the walks enumerated on opposite sides of the equation.