Analog of the cauchy function for a generalized equation with several deviations of the argument

We define a special multiplication of function series (skew multiplication) and a generalized Riemann-Stieltjes integral with function series as integration arguments. The generalized integrals and the skew multiplication are related by an integration by parts formula. The generalized integrals generate a family of linear generalized integral equations, which includes a family (represented in integral form via the Riemann-Stieltjes integral) of linear differential equations with several deviating arguments. A specific feature of these equations is that all deviating functions are defined on the same closed interval and map it into itself. This permits one to avoid specifying the initial functions and imposing any additional constraints on the deviating functions. We present a procedure for constructing the fundamental solution of a generalized integral equation. With respect to the skew multiplication, it is invertible and generates the product of the fundamental solution (a function of one variable) by its inverse function (a function of the second variable). Under certain conditions on the parameters of the equation, the product has all specific properties of the Cauchy function. We introduce the notion of adjoint generalized integral equation, obtain a representation of solutions of the original equation and the adjoint equation in generalized integral Cauchy form, and derive sufficient conditions for the convergence of solutions of a pair of adjoint equations.