Fredholm minors

Definition

Let $a < b$ and let $K \colon [a, b] \times [a, b] \to \C$ be a continuous kernel. For points $z_1, \dots, z_m$ and $w_1, \dots, w_m$ of $[a, b]$, write $$K\begin{pmatrix} z_1, \dots, z_m \\ w_1, \dots, w_m \end{pmatrix} = \det[K(z_i, w_j)]_{i,j=1}^{m},$$ the determinant of the $m \times m$ matrix with entries $K(z_i, w_j)$. Let $n \ge 1$, and let $x_1, \dots, x_n, y_1, \dots, y_n \in [a, b]$. The Fredholm minor of order $n$ of $K$ is the function $$D_n\begin{pmatrix} x_1, \dots, x_n \\ y_1, \dots, y_n \end{pmatrix}(\lambda) = \sum_{k=0}^{\infty} \frac{(-\lambda)^k}{k!} \int_a^b \cdots \int_a^b K\begin{pmatrix} x_1, \dots, x_n, t_1, \dots, t_k \\ y_1, \dots, y_n, t_1, \dots, t_k \end{pmatrix} dt_1 \cdots dt_k$$ of $\lambda \in \C$, where the term with $k = 0$ is $K\begin{pmatrix} x_1, \dots, x_n \\ y_1, \dots, y_n \end{pmatrix}$. The minor of order $1$ is the first minor, written $D(x, y; \lambda)$.