On Lisbon integrals

Abstract

We introduce new complex analytic integral transforms, the Lisbon Integrals, which naturally arise in the study of the affine space $$\mathbb {C}^k$$ of unitary polynomials $$P_s(z)$$ where $$s\in \mathbb {C}^k$$ and $$z\in \mathbb {C}$$ , $$s_i$$ identified to the i-th symmetric function of the roots of $$P_s(z)$$ . We completely determine the $$\mathscr {D}$$ -modules (or systems of partial differential equations) the Lisbon Integrals satisfy and prove that they are their unique global solutions. If we specify a holomorphic function f in the z-variable, our construction induces an integral transform which associates a regular holonomic module quotient of the sub-holonomic module we computed. We illustrate this correspondence in the case of a 1-parameter family of exponentials $$f_t(z) = exp(t z)$$ with t a complex parameter.