Explicit Series and a Certified Hybrid Evaluator for the ${\mathrm{\ell }}_p$ Proximity Operator for $0<p<1$

The nonconvex ${\mathrm{\ell }}_p$ quasi-norm with $0<p<1$ is a powerful sparsity surrogate but makes the proximity operator ${\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{x}}_{\lambda \mathrm{\mid }\cdot {\mathrm{\mid }}^p}$ nontrivial to evaluate robustly. We give an explicit characterization of the scalar proximal map for all $0<p<1$, including the threshold structure and conditions ensuring strict, isolated solutions. Applying the Lagrange–Bürmann inversion to the stationarity equation yields a uniformly convergent series for the larger positive root, which provides an exact and numerically stable formula above the classical threshold. We further derive a Mellin–Barnes (MB) integral representation, explaining its radius of convergence and enabling certified truncation. Building on these ingredients, we design a {certified hybrid evaluator} (short series $+$ truncated vertical MB segment) with a computable a priori error bound that remains accurate in the near-threshold regime. For rational $p$, Gauss’ multiplication formula reduces the coefficients to finite products of shifted Gamma functions, reorganizing the series into a finite sum of generalized hypergeometric functions and explaining the closed forms at $p\in \{1\mathrm{/}3,1\mathrm{/}2,2\mathrm{/}3\}$. We integrate the evaluator into a proximal-gradient method with an inexact proximal oracle and prove convergence under standard summability of the certificates; MATLAB implementations and numerics confirm accuracy, including near-threshold behavior.