  Higher-order non-holomorphic Eisenstein series associated to a Fuchsian group
$\Gamma$ are defined by twisting the series expansion for classical
non-holomorphic Eisenstein series by powers of modular symbols. Their
functional identities include multiplicative and additive factors, making them
distinct from classical Eisenstein series. Here we prove the meromorphic
continuation of these series and establish their functional equations. In
addition, we construct high rank vector bundles $\cal V$ from certain unipotent
representations $\pi$ of $\Gamma$ and show that higher-order non-holomorphic
Eisenstein series can be viewed as components of certain eigensections,
$\mathbb E$, of $\cal V$. With this viewpoint the functional identities of
these higher-order series are formally identical to the classical case. Going
further, we prove bounds for the Fourier coefficients of the higher-order
non-holomorphic Eisenstein series.
