Recursion formulas for the Fourier coefficients of Siegel Eisenstein series of an odd prime level

Abstract: In this paper we treat the Fourier coefficients of Siegel Eisenstein series of level $p$ with trivial or quadratic character, for an odd prime $p$. The Euler $p$-factor of the Fourier coefficient is called the ramified Siegel series. First we show that the ramified Siegel series attached to each cusp can be decomposed to $U(p)$-eigenfunctions explicitly, next we give recursion formulas of such $U(p)$-characteristic ramified Siegel series.