Expanding products of type $\prod_{k=1}^N (1 + \lambda_k x)$

Question

Is there any closed form for the expansion of products of type $\prod_{k=1}^N (1 + \lambda_k x)$ for some sequence $\{\lambda_k\}_{k=1}^N$? In particular, I'm interested in the case where $\lambda_k = k^{-t}$ for some fixed $t > 1$.

Answer 1

Does the following help you (collecting coefficients of $x^n$) $$\prod_{k=1}^N(1+\lambda_kx)=1+\sum_{n=1}^N\sum_{1\leq k_1<\dots<k_n\leq N}\lambda_{k_1}\dots\lambda_{k_n}x^n.$$ Substituting $\lambda_k=k^{-t}$, this becomes $$\prod_{k=1}^N(1+\lambda_kx)=1+\sum_{n=1}^N\sum_{1\leq k_1<\dots<k_n\leq N}\left(k_1\dots k_n\right)^{-t}x^n.$$