Siegel Klingen-Eisenstein series (awaiting review)

Given a modular form $f$ of degree $n_1 \le n$ and weight $k > n + n_1 + 1$ the Klingen-Eisenstein series of $f$ is a modular form $E_{n, n_1}(f)$ of degree $n$ and weight $k$ such that the Siegel $\Phi$ operator maps $E_{n, n_1}(f)$ to $f$.