An explicit computation of the Bergman kernel function
We give an explicit computation of the Bergman kernel function on the domain \(\| z \|^ 2+\| w \|^{2p}<1\) contained in complex Euclidean space of dimension \(n+m\). Here \(p\) is any positive number. We obtain the exact formula \[ K \bigl( (z,w), \overline {(z,w)} \bigr)=\sum^{n+1}_{k=0} c_ k {\bigl( 1-\| z \|^ 2)^{-n- 1+{k \over p}} \over \biggl( \bigl( 1-\| z \|^ 2 \bigr)^{{1 \over p}}-\| w \|^ 2 \biggr)^{m+k}}. \] The constants \(c_ k\) depend on \(k,n,m,p\) and we compute them. Writing \(d_{k+1} = \pi^{n+m} p^ n {c_{k+1} \over \Gamma (m+k+1)}\) we obtain the formula \[ \prod^ n_{l=0} (y+pl)=\sum^ n_{k=-1} d_{k+1} \prod^ k_{l=0} (y+l) \] and determine \(d_ j\) by evaluating the parameter \(y\) at the negative integers. For example \[ d_ 1=-\prod^ n_{l=0} (pl-1)=\prod^ n_{l=1} (pl-1) \quad \text{and } d_ 2={1 \over 2} \prod^ n_{l=0} (pl-2)-\prod^ n_{l=0} (pl-1). \]
scientific article; zbMATH DE number 579773