Berenstein–Zalcman Pizzetti Formulas for the Rank-One Opdam–Cherednik Laplacian

We develop a Pizzetti calculus for the rank-one Opdam–Cherednik setting based on the Weyl symmetrized generalized translation. Its spectral multiplier is the Jacobi function, and the Jacobi differential equation yields a Darboux transmutation between the radial Jacobi operator and the shifted Opdam–Cherednik Laplacian. The classical hypergeometric expansion of the Jacobi multiplier is thereby converted into a Pizzetti series that converges to the symmetrized mean in the Opdam–Cherednik Schwartz topology on each compact spectral band, with explicit shifted factors of Berenstein–Zalcman type. The convergence of the series and the termwise operations are justified using the Opdam–Cherednik Schwartz and Paley–Wiener theories. We also obtain mean-value and finite shifted-polyharmonic characterizations, a normalized ball-mean expansion, and a quantitative two-radius estimate. The even specialization recovers the known radial Laplace–Beltrami calculus on rank-one noncompact symmetric spaces, while a precise curvature contraction yields the rank-one rational Dunkl Pizzetti formula. The aspect specific to the present formulation is that these identities are established for a reflection–differential spatial operator before restriction to the even sector; no novelty is claimed for the underlying Jacobi expansion.