A combinatorial correspondence for walks in Weyl chambers

The m × m determinant of hyperbolic Bessel functions det |Iai − bj(2x)| can be factored into two smaller determinants by elementary operations if ai = −am + 1 − i and bi = −bm + 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 finding a sequence of reflections which give a correspondence between the walks enumerated on opposite sides of the equation.