Weighted sum of the extensions of the representations of quadratic forms

Let L, N and M be positive definite integral ({mathbb{Z}}) -lattices. In this paper, we show some relation between the weighted sum of representations of L and N by gen(M) and the weighted sum of extensions of (tilde M_{tilde sigma}) in the gen(M _σ) via N _η when M is even and gcd(dL, dM) =  1. As a consequence of the particular case when M is even unimodular, we recapture the Böcherer formula (13) in (Böcherer, Maths Z 183:21–46, 1983) for the relation of the Fourier coefficients between Eisenstein series and Jacobi–Eisenstein series.