Consider a finite complex $E$ of (holomorphic) vector bundles on a (complex) manifold $X$, i.e, the complex is of the form $$ 0 \to E_N \to E_{N-1} \to \dots \to {E_0} \to 0, $$ where the bundles are equipped with connections $D_i$. By K-theory, one may consider the complex $E$ as the alternating sum $\sum_i (-1)^i [E_i]$, and it is then natural to define the Chern character (as a form) as $ch(E,D) := \sum_{i=0}^N (-1)^i ch(E_i,D_i)$, and the Chern form as $c(E,D) := \prod_{i=0}^N c(E_i,D_i)^{(-1)^i}$, where $ch(E_i,D)$ and $c(E_i,D)$ denote the Chern character and Chern form of $(E_i,D_i)$.

Alternatively, for a fixed $k$, one may express the Chern character as a polynomial in the Chern forms, $ch_k = S_k(c_1,\dots,c_k)/k!$, where $S_k$ is the polynomial which expresses the Newton polynomials in terms of the elementary symmetric polynomials, i.e., what is sometimes called the Hirzebruch-Newton polynomial. For example, $S_1(t_1)=t_1$, $S_2(t_1,t_2)=t_1^2-2t_2$ etc. With the help of the polynomials $S_k$ above, one could alternatively define a Chern character of $E$ by $\widetilde{ch}_k(E,D)=S_k(c_1(E,D),\dots,c_k(E,D))/k!$.

I have only found mentioned in passing or implicitly that these definitions coincide, i.e., $ch_k(E,D) = \widetilde{ch}_k(E,D)$, but not any precise argument. Does anyone know of a convenient reference or proof of this fact?