I have an expression, in which I am looking for a formula to calculate
$$ \prod_{n=0}^N (a^n +b)$$
as a summation. Is there a general name for this type of formula? Or a relevant approach to find partial products such as this one?
2026-04-03 08:03:08.1775203388
Find a formula for calculate $\prod_{n=0}^N(a^n + b)$
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Using the Gaussian binomial coefficients, also known as the q-analogs of the binomial coefficient, gives a representation of: $$ \prod_{n=0}^{N} (a^n + b) = b^{N+1} \: \left(-\frac{1}{b}; \: a \right)_{N+1} = b^{N+1} \: \sum_{j=0}^{N+1} {{N+1} \brack {j}}_a \: \left( \frac{1}{b}\right)^j a^{\binom{j}{2}} $$ Though I think that sum would be much more computationally expensive than just running the product through.