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There seems to be a result for formal series (I hope this is right) for all integer $r\ge 0$ $$ \sum_{n\ge 0} (-x)^n\ {{n+r}\choose{r}}_{q} = (1+x)^{-1}(1+qx)^{-1}\dots (1+q^{r}x)^{-1} $$ where the $q$...

It is going to be a rather long question, so I will first state it and then try to explain and motivate it. Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...