I came cross the following serie :
$$\sum\limits_{\mathbf k \in \mathbb N^d} e^{\langle \mathbf r, \mathbf k \rangle}$$
What would be the conditions on the d-dimensional real vector $\mathbf r$ for the convergence of this serie ? What would be the obtained value of the serie in this case ?
More generally, is there any litterature / a name / some referenc ei could read about this sort of multivariate power serie ?