An explicit computation of the Bergman kernel function

Abstract

We give an explicit computation of the Bergman kernel function on the domain

$$||z||^2 + ||w||^{2p}< 1$$

contained in complex Euclidean space of dimensionn +m. Herep is any positive number. We obtain the exact formula

$$K((z,w), (\overline {z,w)} ) = \sum\limits_{k = 0}^{n + 1} {^c k\frac{{(1 - ||z||^2 )^{ - n - 1 + \frac{k}{p}} }}{{((1 - ||z||^2 )^{\frac{1}{p}} - ||w||^2 )^{m + k} }}.} $$

The constants c_k depend onk, n, m, p and we compute them. Writing$d_{k + 1} = \pi ^{n + m} p^n \frac{{^c k + 1}}{{\Gamma (m + k + 1)}}$ we obtain the formula

$$\prod\limits_{l = 0}^n {(y + pl)} = \sum\limits_{k = - 1}^n {d_{k + 1} } \prod\limits_{l = 0}^k {(y + l)} $$

and determined _j by evaluating the parametery at the negative integers. For example

$$d_1 = - \prod\limits_{l = 0}^n {(pl - 1)} = \prod\limits_{l = 1}^n {(pl - 1) and } d_2 = \frac{1}{2}\prod\limits_{l = 0}^n {(pl - 2)} - \prod\limits_{l = 0}^n {(pl - 1)} .$$