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

R. Dmytryshyn; C. Cesarano; I.-A. Lutsiv

doi:10.15421/242515

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

R. Dmytryshyn (Vasyl Stefanyk Precarpathian National University), https://orcid.org/0000-0003-2845-0137
C. Cesarano (International Telematic University UNINETTUNO), https://orcid.org/0000-0002-1694-7907
I.-A. Lutsiv (Vasyl Stefanyk Precarpathian National University), https://orcid.org/0000-0002-4100-2972

Abstract

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.

Keywords

hypergeometric function; branched continued fraction; analytic function; approximation by rational functions; convergence; analytic continuation

MSC 2020

Pri 33C65, Sec 32A17, 41A20, 32A10, 40A99, 30B40

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