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I would like to compute the coefficients (in $y$) of the following series $$\prod_{i=1}^n\frac{x_i+y}{1-e^{-(x_i+y)}}$$ Is there a way to extract them using residue calculation?

BinAcker
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  • What can you say about the $y_i$ ? Positive ? – Jean Marie Jan 30 '22 at 13:54
  • @JeanMarie the $x_i$ are non-zero. To say the truth for me both the $x_i$ and $y$ are cohomology classes (Chern classes of line bundles) and I am interested in integrating that sum where only some power of $y$ contribute (for example https://math.stackexchange.com/questions/4366639/residue-computation-for-hirzebruch-riemann-roch ). Hence I am interested in extracting the coefficient of a specific power of $y$ – BinAcker Jan 30 '22 at 14:00
  • But is it the coefficient of any power $y^n$ ? I mean possibly very large values of $n$ ? Because otherwise any Computer Algebra System will give you a result till - say - the coefficient of term $y^{10}$ by expanding the product of formal series $\Pi f(x_i+x)$ where $f(x):=x/(1-e^{-x})$. – Jean Marie Jan 30 '22 at 14:18
  • Would you be interested by a numerical approach like in my answer here ? – Jean Marie Jan 30 '22 at 14:52
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    That is interesting but I'd rather have some hands on approach :), this came out while I was trying to prove by hand some special case of the Grothendieck-Riemann-Roch theorem, in other term I know what the answer should be, I just want to retrieve it. – BinAcker Jan 30 '22 at 15:51
  • You mean you want to have an exact expression for the residue, but what would you take as circuit $\gamma$, but without any pole as far as I know. – Jean Marie Jan 30 '22 at 16:06
  • ... because $f(x)=x/(1-\exp(-x))$ (with $f(0):=1$) has no pole... therefore no computable residue. – Jean Marie Jan 30 '22 at 19:20
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    @JeanMarie I asked the full question on overflow https://mathoverflow.net/questions/415055/proving-by-hand-grothendieck-riemann-roch-for-mathcalo-e1-to-mathbbpe – BinAcker Feb 01 '22 at 12:56
  • What is the expected answer using GRR? Also, for your particular example, are you interested in all the coefficients of $y$ or in the one of degree $n$? – Aitor Iribar Lopez Feb 01 '22 at 20:42

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