On $p$-adic Siegel Eisenstein series from a point of view of the theory of mod $p^m$ singular forms

We show that the $p$-adic Siegel Eisenstein series of general degree attached to two kind of number sequences are both linear combinations of genus theta series of level $p$, by applying the theory of mod $p$-power singular forms. As special cases of this result, we derive the result of Nagaoka and Katsurada–Nagaoka.