Re: Question on finding eta expansions

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Fri, Sep 13, 2024 at 07:22:49PM +0000, LNU, Swati wrote:
> Hello,
> I am trying to find the expansion of the eta quotient f(z) = \eta(7z)
> \eta(z)^2 in fractional powers of q^(1/24) and I tried the following code:
> D(n) = {q^(9/24) * eta(q^7 + O(q^(n+1))) * eta(q + O(q^(n+1)))^(2) ;}

PARI just cannot define power series with fractional exponents.

You have two options:
- Do the multiplication by q^(9/24) 'in you head'
D(n) = { eta(q^7 + O(q^(n+1))) * eta(q + O(q^(n+1)))^(2) ;}
? D(10)
%6 = 1-2*q-q^2+2*q^3+q^4+2*q^5-2*q^6-q^7-q^9-q^10+O(q^11)

Set Q = q^(1/24) (in your head) and then do
D(n) = {Q^9 * eta(Q^(24*7) + O(Q^(24*(n+1)))) * eta(Q^24 + O(Q^(24*(n+1))))^(2) ;}

? D(10)
%4 = Q^9-2*Q^33-Q^57+2*Q^81+Q^105+2*Q^129-2*Q^153-Q^177-Q^225-Q^249+O(Q^273)

For exampke the second exponent is 9/24 + 1 = 33/24

Cheers,
Bill.