Algebraic independence of certain almost polyadic series

We study the arithmetic properties of almost polyadic numbers
$$\sum_{n=1}^\infty a_{i}\left(a_{i}+b_{i}\right)\ldots\left(a_{i}+\left(n-1\right)b_{i}\right),i=1,...,m,$$
where the numbers $a_{i},b_{i}\in\mathbb Z$, $\left(a_{i},b_{i}\right)=1$.
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