  We investigate a class of parametric elliptic eigenvalue problems with
homogeneous essential boundary conditions where the coefficients (and hence the
solution $u$) may depend on a parameter $y$. For the efficient approximate
evaluation of parameter sensitivities of the first eigenpairs on the entire
parameter space we propose and analyse Gevrey class and analytic regularity of
the solution with respect to the parameters. This is made possible by a novel
proof technique which we introduce and demonstrate in this paper. Our
regularity result has immediate implications for convergence of various
numerical schemes for parametric elliptic eigenvalue problems, in particular,
for elliptic eigenvalue problems with infinitely many parameters arising from
elliptic differential operators with random coefficients.
