With a given measure \(m(t)dt\) on \(\mathbb{R}^+\), associate its moments \(\gamma_n= \int^\infty_0 t^nm(t) dt\) \((n= 0,1,2,\dots)\), its Bergman kernel function \(K_m(t)= \sum_n \gamma^{-1}_n t^n\), the new function \(\widetilde m(t)= K_m(t)^{-1}m(t)\), and the corresponding kernel function \(K_{\widetilde m}(t)\). The model case is \(m(t)= e^{-t}\): then \(\gamma_n(t)= n!\) and (A) \(K_m(t)m(t)= 1\), (B) \(K_{\widetilde m}(t)= 2K^2_m(t)\), and (C) \(\iint_{\mathbb{C}}|K_{\widetilde m}(\overline az)|m(|z|^2) dx dy= 2\pi K_m(|a|^2)\). The main goal of this paper is to prove that for a large class of measures \(m(t) dt\), the quantities \(\gamma_n\), \(K_m\) and \(K_{\widetilde m}\) satisfy asymptotic relations similar to the exact relations (A), (B), (C). The corresponding theorems and proofs have substantial technical details, but there is a section which has informal statements and proof outlines of how the asymptotic growth of \((\gamma_n)\) is controlled by the growth of \(m\), and of how the growth of \(K_m(re^{it})\) for large \(r\) is controlled by the growth of the coefficient sequence of a certain entire function.