Negative-index scalar Riemann-Hilbert moment conditions (${\int }_L{t}^{j}g(t)dt=0(0\le j<q)$)

Suppose a scalar canonical factor has exterior growth $X_{-}(z)\sim z^q$ with index $-q<0$, and the original exterior unknown must decay as $O(z^{-1})$. Dividing by the canonical factor gives an additive jump $D_{+}-D_{-}=g$ with exterior decay $O(z^{-q-1})$. The exterior Cauchy integral expands in moments $-{\int }_L{t}^{j}g(t)dt/(2\pi i{z}^{j+1})$. Its first $q$ coefficients must therefore vanish, yielding the displayed solvability conditions. Under this decay there is no nonzero entire polynomial addition. A nonlocal moment appearing in $g$ must additionally agree with the moment of the reconstructed original jump.