The Green function for the weighted biharmonic operator $\Delta(1-|z|^2)^{-\alpha}\Delta$, and factorization of analytic functions

An explicit integral formula is obtained for the Green function of the weighted biharmonic operator $\Delta(1-|z|^2)^{-\alpha}\Delta$ in the unit disc of the complex plane for the case $\alpha\in(-1,0)$. The formula shows the positivity of the Green function. This is a basis for a theorem on factorization of analytic functions in the weighted Bergman spaces with the weights $w(z)=(1-|z|^2)^\alpha$ as product of a nonvanishing function and a function of special form responsible for the zeros. Bibliography: 16 titles.