An explicit computation of the Bergman kernel function

We give an explicit computation of the Bergman kernel function on the domain % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaacY% hacaWG6bGaaiiFaiaacYhadaahaaWcbeqaaiaaikdaaaGccqGHRaWk% caGG8bGaaiiFamXvP5wqonvsaeHbfv3ySLgzaGqbciab-Dha3jab-X% ha8jab-Xha8naaCaaaleqabaacbaGaa4NmaGqaciaa9bhaaaGccqGH% 8aapcaaIXaaaaa!4BC1! $$||z||^2 + ||w||^{2p}< 1$$ contained in complex Euclidean space of dimensionn +m. Herep is any positive number. We obtain the exact formula % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saKazaa% iacaGGOaqcaaMaaiikaiaadQhacaGGSaWexLMBb50ujbqeguuDJXwA% KbacfiGae83DaChcbaGaa4xkaiaa+XcaruqqYLwySbacgaGaa0hiai% aa+HcakmaanaaabaqcaaMaamOEaiaacYcacqWF3bWDcaGFPaaaaKaz% aaiacaGGPaGaeyypa0JcdaaeWbqaamaaCaaaleqabaGai4mGdogaaa% GccaWGRbWaaSaaaeaacaGGOaGaaGymaiabgkHiTiaacYhacaGG8bGa% amOEaiaacYhacaGG8bWaaWbaaSqabeaacaaIYaaaaOGaaiykamaaCa% aaleqabaGaeyOeI0IaamOBaiabgkHiTiaaigdacqGHRaWkdaWcaaqa% aiaadUgaaeaacaWGWbaaaaaaaOqaaKazaaiacaGGOaqcaaMaaiikaO% GaaGymaiabgkHiTiaacYhacaGG8bGaamOEaiaacYhacaGG8bWaaWba% aSqabeaacaaIYaaaaKaaGjaa+LcakmaaCaaaleqabaWaaSaaaeaaca% aIXaaabaGaamiCaaaaaaGccqGHsislcaGG8bGaaiiFaiab-Dha3jaa% cYhacaGG8bWaaWbaaSqabeaacaaIYaaaaKazaaiacaGGPaGcdaahaa% Wcbeqaaiaad2gacqGHRaWkcaWGRbaaaaaakiaac6caaSqaaiaadUga% cqGH9aqpcaaIWaaabaGaamOBaiabgUcaRiaaigdaa0GaeyyeIuoaaa% a!8195! $$K((z,w), (\overline {z,w)} ) = \sum\limits_{k = 0}^{n + 1} {^c k\frac{{(1 - ||z||^2 )^{ - n - 1 + \frac{k}{p}} }}{{((1 - ||z||^2 )^{\frac{1}{p}} - ||w||^2 )^{m + k} }}.} $$ The constants ck depend onk, n, m, p and we compute them. Writing % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa% aaleaacaWGRbGaey4kaSIaaGymaaqabaGccqGH9aqpcqaHapaCdaah% aaWcbeqaaiaad6gacqGHRaWkcaWGTbaaaOGaamiCamaaCaaaleqaba% GaamOBaaaakmaalaaabaWaaWbaaSqabeaacGaYao4yaaaakiaadUga% cqGHRaWkcaaIXaaabaGaeu4KdCKaaiikaiaad2gacqGHRaWkcaWGRb% Gaey4kaSIaaGymaiaacMcaaaaaaa!4DA7! $$d_{k + 1} = \pi ^{n + m} p^n \frac{{^c k + 1}}{{\Gamma (m + k + 1)}}$$ we obtain the formula % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebCaeaaca% GGOaGaamyEaiabgUcaRiaadchacaWGSbGaaiykaaWcbaGaamiBaiab% g2da9iaaicdaaeaacaWGUbaaniabg+GivdGccqGH9aqpdaaeWbqaai% aadsgadaWgaaWcbaGaam4AaiabgUcaRiaaigdaaeqaaaqaaiaadUga% cqGH9aqpcqGHsislcaaIXaaabaGaamOBaaqdcqGHris5aOWaaebCae% aacaGGOaGaamyEaiabgUcaRiaadYgacaGGPaaaleaacaWGSbGaeyyp% a0JaaGimaaqaaiaadUgaa0Gaey4dIunaaaa!5661! $$\prod\limits_{l = 0}^n {(y + pl)} = \sum\limits_{k = - 1}^n {d_{k + 1} } \prod\limits_{l = 0}^k {(y + l)} $$ and determined j by evaluating the parametery at the negative integers. For example % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa% aaleaacaaIXaaabeaakiabg2da9iabgkHiTmaarahabaGaaiikaiaa% dchacaWGSbGaeyOeI0IaaGymaiaacMcaaSqaaiaadYgacqGH9aqpca% aIWaaabaGaamOBaaqdcqGHpis1aOGaeyypa0ZaaebCaeaacaGGOaGa% amiCaiaadYgacqGHsislcaaIXaGaaiykaerbbjxAHXgaiuaacaWFGa% Gaa8hiaiaa-bcacaWFGaacbaGaa4xyaiaa+5gacaGFKbGaa8hiaiaa% -bcacaWFGaaaleaacaWGSbGaeyypa0JaaGymaaqaaiaad6gaa0Gaey% 4dIunakiaadsgadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaWcaaqa% aiaaigdaaeaacaaIYaaaamaarahabaGaaiikaiaadchacaWGSbGaey% OeI0IaaGOmaiaacMcaaSqaaiaadYgacqGH9aqpcaaIWaaabaGaamOB% aaqdcqGHpis1aOGaeyOeI0YaaebCaeaacaGGOaGaamiCaiaadYgacq% GHsislcaaIXaGaaiykaaWcbaGaamiBaiabg2da9iaaicdaaeaacaWG% Ubaaniabg+GivdGccaGGUaaaaa!7510! $$d_1 = - \prod\limits_{l = 0}^n {(pl - 1)} = \prod\limits_{l = 1}^n {(pl - 1) and } d_2 = \frac{1}{2}\prod\limits_{l = 0}^n {(pl - 2)} - \prod\limits_{l = 0}^n {(pl - 1)} .$$