On the convergence of polynomial Fredholm series

In this note, we study the infinite system of Fredholm series of polynomials in $\lambda$, formed, in the classical way, for a kernel on $\mathbb{R}^2$ of the form $\boldsymbol{H}(s,t)-\lambda\boldsymbol{S}(s,t)$, where $\lambda$ is a complex parameter. We establish a convergence of these series in the complex plane with respect to sup-norms of various spaces of continuous functions. The convergence results apply to solving a Fredholm integral equation with a kernel that is linear with respect to parameter.