  Due to their singularities, multiple zeta functions behave sensitively at
non-positive integer points. In this article, we focus on the asymptotic
behavior at the origin $(0,\dots, 0)$ and unveil the generating series of the
asymptotic coefficients as a generalization of the classical Gregory
coefficients. This enables us to reveal the underlying symmetry of the
asymptotic coefficients. Additionally, we extend the relationship between the
asymptotic coefficients and the Gregory coefficients to include Hurwitz
multiple zeta functions.
