Sage function with functionality of series command from Maple.

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 7 years ago

dan_fulea
5650 ●5 ●44 ●97

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

link

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 ( 7 years ago )

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Sage function with functionality of series command from Maple.

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Sage function with functionality of series command from Maple.