Computational Aspects of Approximating the Horn Hypergeometric Functions \(H_3\) by Branched Continued Fractions

This paper investigates the approximation of Horn hypergeometric function \(H_3\) using branched continued fractions (BCFs).Based on the formal branched continued fraction expansion for the ratio of hypergeometric functions \(H_3\), a branched continued fraction expansion for a specific function is constructed. Numerical experiments using a custom Python implementation compare the convergence properties of the BCF approximants with the partial sums of the corresponding double power series. Results, presented in tables and plots, demonstrate that the BCF approach generally offers better convergence properties, including potentially wider regions of convergence and higher accuracy, particularly in regions where the power series diverges or converges slowly. The convergence behavior is visualized through error plots in different complex planes, suggesting that theBCF provides a robust tool for approximating this special function. Additionally, algorithms for computing approximants of continued fractions are studied. The results show that the continuant method is unstable and slower than the BR algorithms. The BR algorithms are stable, and their parallel implementation is faster than the single-threaded version.