This paper explores the representation and analytical continuation of Lauricella–Saran’s hypergeometric function \( F_K \) using branched continued fractions. The authors establish new symmetric domains for the analytical continuation of \( F_K \) under certain conditions on real and complex parameters. They use a technique to extend the convergence from a small domain to a larger one and apply the PC method to prove that these domains are also domains of analytical continuation. The paper includes definitions and preliminary results on branched continued fractions, and provides detailed proofs for several theorems and corollaries. The main findings include the convergence of branched continued fractions in specific domains and the analytic continuation of the function \( F_K \) within these domains. The authors also discuss the applicability of these results to other special functions and highlight the importance of parabolic regions in forming cardioid domains.This paper explores the representation and analytical continuation of Lauricella–Saran’s hypergeometric function \( F_K \) using branched continued fractions. The authors establish new symmetric domains for the analytical continuation of \( F_K \) under certain conditions on real and complex parameters. They use a technique to extend the convergence from a small domain to a larger one and apply the PC method to prove that these domains are also domains of analytical continuation. The paper includes definitions and preliminary results on branched continued fractions, and provides detailed proofs for several theorems and corollaries. The main findings include the convergence of branched continued fractions in specific domains and the analytic continuation of the function \( F_K \) within these domains. The authors also discuss the applicability of these results to other special functions and highlight the importance of parabolic regions in forming cardioid domains.