On the analytical continuation of the ratio $H_4(\alpha,\delta+1;\gamma,\delta;-\mathbf{z})/H_4(\alpha,\delta+2;\gamma,\delta+1;-\mathbf{z})$

The paper considers the problem of analytical continuation of special functions by branched continued fractions. These representations play an important role in approximating of special functions that arise in various applied problems. By improving the methods of studying the convergence of branched continued fractions, several domains of analytical continuation of the special function $H_4(\alpha,\delta+1;\gamma,\delta;-\mathbf{z})/H_4(\alpha,\delta+2;\gamma,\delta+1;-\mathbf{z})$ in the case of real and complex parameters are established. To prove the analytical continuation, the so-called PC method is used, which is based on the principle of correspondence between a formal double power series and a branched continued fraction. An example is provided at the end.