It is a well-known result that if a nonconstant meromorphic function $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f$ on $\mathbb{C}$ and its $l$th derivative $f^{(l)}$ have no zeros for some $l\geq 2$, then $f$ is of the form $f(z)=\exp (Az+B)$ or $f(z)=(Az+B)^{-n}$ for some constants $A$, $B$. We extend this result to meromorphic functions of several variables, by first extending the classic Tumura–Clunie theorem for meromorphic functions of one complex variable to that of meromorphic functions of several complex variables using Nevanlinna theory.

We investigate the multivariate sampling theory associated with multiparameter eigenvalue problems. A several-variable counterpart of the classical sampling theorem of Whittaker, Kotel’nikov and Shannon is given. It arose when the multiparameter system has order one. Two-dimensional sampling theorems associated with two-parameter systems of second-order differential operators will be established. The sampling formulae are of multivariate non-uniform Lagrange interpolation type. Unlike many of the known formulae, the interpolating functions are not necessarily products of single variable functions.