Representation of the Stieltjes Integral in Terms of the Riemann Integral Depending on a Parameter

In this paper, we obtain a new formula for the representation of the Riemann–Stieltjes integral of a continuous function in terms of the passage to the limit with respect to the parameter in a Riemann integral depending on this parameter. The derivation of this formula is based on the study of the functional properties of the solution of the auxiliary difference equation of first order representing the weighted first difference of a given function in the form of a simple first difference of an unknown function. The result obtained can be used for the analytic and approximate calculation of Stieltjes integrals.