The Gauss continued fraction for the ratio of two hypergeometric functions is converted into an ordinary fraction (all partial numerators are 1) and simplifications occurring for particular relations between the parameters are discussed. In particular, a very simple expansion is obtained for the ratio ${E /K}$ of the complete elliptic integrals. For the argument $ - z$ and for certain ranges of the parameters, the Gauss expansion is a Stieltjes fraction and represents the input impedance of a passive ladder network. The Stieltjes integral representations of the corresponding positive real functions are established and yield many new definite integrals. A general method for obtaining indefinite integrals involving independent solutions of a self-adjoins differential equation, by means of the Wronskian, is also mentioned. Finally, some continued fractions originating from other contiguity relations for hypergeometric functions are discussed.