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Let $\phi\in C^\infty(\Complex^n)$ be a given real valued function. We assumethat $\pr\ddbar\phi$ is non-degenerate of constant signature $(n_-,n_+)$ on$\Complex^n$. When $q=n_-$, it is well-known that the Bergman kernel for$(0,q)$ forms with respect to the $k$-th weight $e^{-2k\phi}$, $k>0$, admits afull asymptotic expansion in $k$. In this paper, we compute the trace of thesecond coefficient of the asymptotic expansion on the diagonal.

The second coefficient of the asymptotic expansion of the weighted Bergman kernel for $(0,q)$ forms on $\Complex^n$