A determinant identity for the sum of contour integral matrices

Abstract

We derive an identity for the determinant of the sum of two $n\times n$ matrices, $A$ and $B$, whose entries are defined via contour integrals. Specifically, we consider $A(i,j)=\frac{1}{2\pi\mathrm{i}}\oint_0 z^{i-j-1}p_i(z)f_j(z)\mathrm{d} z$ and $B(i,j)= \frac{1}{2\pi\mathrm{i}}\int_{\Gamma} q_i(z)g_j(z) \mathrm{d} z$. Under suitable assumptions on the functions $p,q,f,g$, we show that $\det(A+B)$ can be expressed as a Fredholm determinant $\det(\mathrm{I} +K)$, where $K$ is an integral kernel acting on the contour $\Gamma$. This result generalizes a recent identity obtained in \cite{Baik-Liao-Liu26}.