The paper by Dmytryshyn R., Lutsiv I.-A., and Dmytryshyn M. explores the analytic extension of Horn’s hypergeometric function \( H_4 \) using branched continued fraction expansions. The authors establish new convergence domains for these expansions, which enable the analytical continuation of the function to these domains. The key results include: 1. **Theorem 2**: Provides convergence criteria for complex values of \( h_k \), where \( h_k \) are defined by the formula \( h_k = \frac{(2c-a+k-1)(a+k)}{(c+k-1)(c+k)} \). 2. **Theorem 3**: Establishes convergence for negative values of \( h_k \). 3. **Theorem 4**: Establishes convergence for positive values of \( h_k \). These theorems provide specific domains where the branched continued fraction expansions converge uniformly, allowing for the analytical continuation of \( H_4 \) in these regions. The paper also includes examples to illustrate the application of these results. The authors conclude by discussing potential future research directions, such as extending the domains of convergence using parabolic and angular domains, analyzing truncation errors, and studying computational stability.The paper by Dmytryshyn R., Lutsiv I.-A., and Dmytryshyn M. explores the analytic extension of Horn’s hypergeometric function \( H_4 \) using branched continued fraction expansions. The authors establish new convergence domains for these expansions, which enable the analytical continuation of the function to these domains. The key results include: 1. **Theorem 2**: Provides convergence criteria for complex values of \( h_k \), where \( h_k \) are defined by the formula \( h_k = \frac{(2c-a+k-1)(a+k)}{(c+k-1)(c+k)} \). 2. **Theorem 3**: Establishes convergence for negative values of \( h_k \). 3. **Theorem 4**: Establishes convergence for positive values of \( h_k \). These theorems provide specific domains where the branched continued fraction expansions converge uniformly, allowing for the analytical continuation of \( H_4 \) in these regions. The paper also includes examples to illustrate the application of these results. The authors conclude by discussing potential future research directions, such as extending the domains of convergence using parabolic and angular domains, analyzing truncation errors, and studying computational stability.