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We consider a linear Volterra integral equation of the second kind with a sumkernel $K(t',t)=\sum_i K_i(t',t)$ and give the solution of the equation interms of solutions of the separate equations with kernels $K_i$, provided theseexist. As a corollary, we obtain a novel series representation for the solutionwith improved convergence properties. We illustrate our results with examples,including the first known Volterra equation solved by Heun's confluentfunctions. This solves a long-standing problem pertaining to the representationof such functions. The approach presented here has widespread applicability inphysics via Volterra equations with degenerate kernels.

On the solutions of linear Volterra equations of the second kind with sum kernels