Abstract. We investigate the Berezin integral of non-compactly supported quantities. In the framework of supermanifolds with corners, we give a general, explicit and coordinate-free repesentation of the boundary terms introduced by an arbitrary change of variables. As a corollary, a general Stokes's theorem is derived-here, the boundary integral contains transversal derivatives of arbitrarily high order.

We compute the Harish-Chandra c-function for a generic class of rank-one purely non-compact Riemannian symmetric superspaces X = G/K in terms of Euler Γ functions, proving that it is meromorphic. Compared to the even case, the poles of the c-function are shifted into the right half-space. We derive the full asymptotic Harish-Chandra series expansion of the spherical superfunctions on X. In the case where the multiplicity of the simple root is an even negative number, they have a closed expression as Jacobi polynomials for an unusual choice of parameters.

We compute the Harish-Chandra c-function for a generic class of rank-one purely non-compact Riemannian symmetric superspaces X = G/K in terms of Euler Γ functions, proving that it is meromorphic. Compared to the even case, the poles of the c-function are shifted into the right half-space. We derive the full asymptotic Harish-Chandra series expansion of the spherical superfunctions on X. In the case where the multiplicity of the simple root is an even negative number, they have a closed expression as Jacobi polynomials for an unusual choice of parameters.

Abstract. For the super-hyperbolic space in any dimension, we introduce the non-Euclidean Helgason-Fourier transform. We prove an inversion formula exhibiting residue contributions at the poles of the Harish-Chandra c-function, signalling discrete parts in the spectrum. The proof is based on a detailed study of the spherical superfunctions, using recursion relations and localization techniques to normalize them precisely, careful estimates of their derivatives, and a rigorous analysis of the boundary terms appearing in the polar coordinate expression of the invariant integral.