Hybrid Algorithm via Reciprocal-Argument Transformation for Efficient Gauss Hypergeometric Evaluation in Wireless Networks

The rapid densification of wireless networks demands efficient evaluation of special functions underpinning system-level performance metrics. To facilitate research, we introduce a computational framework tailored for the zero-balanced Gauss hypergeometric function Ψ ( x , y ) ≜ F 1 2 ( 1 , x ; 1 + x ; − y ) , a fundamental mathematical kernel emerging in Signal-to-Interference-plus-Noise Ratio (SINR) coverage analysis of non-uniform cellular deployments. Specifically, we propose a novel Reciprocal-Argument Transformation Algorithm (RTA), derived rigorously from a Mellin–Barnes reciprocal-argument identity, achieving geometric convergence with O 1 / y . By integrating RTA with a Pfaff-series solver into a hybrid algorithm guided by a golden-ratio switching criterion, our approach ensures optimal efficiency and numerical stability. Comprehensive validation demonstrates that the hybrid algorithm reliably attains machine-precision accuracy ( ∼ 10 − 16 ) within 1 μs per evaluation, dramatically accelerating calculations in realistic scenarios from hours to fractions of a second. Consequently, our method significantly enhances the feasibility of tractable optimization in ultra-dense non-uniform cellular networks, bridging the computational gap in large-scale wireless performance modeling.