Sage function with functionality of series command from Maple.

asked 2018-01-18 01:07:38 +0100

Anonymous

Hello all,

I'm relatively new to Sage and I'm currently attempting to compute the truncation of some infinite products. For example I want to compute the coefficients of $$ \frac{\eta(z)}{\eta(4z)\eta(2z)} $$ as an infinite series, where $$ \eta(z) = q^{\frac{1}{24}}\prod_{n = 1}^{\infty}(1 - q^{n}) $$. Maple has a function called series that does exactly what I want. I would've linked it but I don't have enough karma. If you want to find the function just google "series command maple" and click the first link. I was wondering if Sage has a similar function. Thanks!

answered 2018-01-18 10:34:49 +0100

dan_fulea
5537 ●5 ●43 ●95

Up to the multiplcative factor traced back from $q^{1/24}$, there is the following expansion:

sage: PSR.<q> = PowerSeriesRing( QQ, default_prec=20 )
sage: f = qexp_eta( PSR, 20 )
sage: f
1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + O(q^20)
sage: f(q) / f(q^2) / f(q^4)
1 - q - q^3 + 2*q^4 - 2*q^5 + q^6 - 2*q^7 + 5*q^8 - 5*q^9 + 3*q^10 - 5*q^11 + 10*q^12 - 10*q^13 + 7*q^14 - 11*q^15 + 20*q^16 - 20*q^17 + 15*q^18 - 22*q^19 + O(q^20)

Note: ?qexp_eta describes the used function.

Note: See also ?EtaProduct - EtaProduct is already implemented for the case where the product has "an integer power of $q$ cumulated from the many factors". Here is the link: etaproducts.html

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Comments

sage: eta? Signature: eta(x) Docstring:
Return the value of the eta function at "x", which must be in the upper half plane.

   The eta function is

      eta(z) = e^{pi i z / 12} prod_{n=1}^{infty}(1-e^{2pi inz})

Where's the beef ?

Emmanuel Charpentier ( 2018-01-20 09:45:36 +0100 )edit

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Sage function with functionality of series command from Maple.