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In this paper, we study an infinite system of Fredholm series of polynomialsin $\lambda$, formed, in the classical way, for a continuous Hilbert-Schmidtkernel on $\mathbb{R}\times\mathbb{R}$ of the form$\boldsymbol{H}(s,t)-\lambda\boldsymbol{S}(s,t)$, where $\lambda$ is a complexparameter. We prove a convergence of these series in the complex plane withrespect to sup-norms of various spaces of continuous functions vanishing atinfinity. The convergence results enable us to solve explicitly an integralequation of the second kind in $L^2(\mathbb{R})$, whose kernel is of the aboveform, by mimicking the classical Fredholm-determinant method.

On Fredholm's Integral Equations on the Real Line, Whose Kernels Are Linear in a Parameter