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The classical Galton--Watson process works with a fixed probability offission at each time step. One of the generalizations is that the probabilitiesdepend on time. We consider one of the most complex and interesting cases whenwe do not know the exact probabilities of fission at each time step - theseprobabilities are random variables themselves. The limit distributions of thenumber of descendants are described in terms of generalized integral anddifferential functional equations of the Schr\"oder type. There are no moreanalogs of periodic Karlin-McGregor functions, which were very helpful in theanalysis of the asymptotic behavior of limit distributions for the classicalcase. We propose some approximate asymptotic methods. Even simple cases ofrandom families with one or two members lead to nice asymptotics involving,interesting problems related to special functions and special constants. One ofthem, Example 2 is already announced on\href{https://math.stackexchange.com/questions/4748129/asymptotics-of-sequence-of-rational-numbers}{this}and\href{https://mathoverflow.net/questions/458885/simple-integral-equation}{this}sites. Finally, the phenomenon of why the oscillations in the main asymptoticterm usual for the classical case become rare in the case of randomenvironments is discussed.

Generalized Schr\"oder-type functional equations for Galton-Watson processes in random environments