How to expand Lie characters?

Question

The following involves characters of affine Lie algebras, and I will be using as reference the book on CFT by Francesco et at (here are some screenshots if useful). But hopefully the post will be self-contained.

Let $$ \Theta^k_\lambda=\sum_{n\in\mathbb Z}\exp\big[-2\pi i(knz+\frac12\lambda z-kn^2\tau-n\lambda\tau-\lambda^2\tau/4k)\big]\tag{14.176} $$ and $$ \chi^k_\lambda=\frac{\Theta^{k+2}_{\lambda+1}-\Theta^{k+2}_{-\lambda-1}}{\Theta^2_1-\Theta^2_{-1}}\tag{14.174} $$

The idea is to expand $\chi$ for $\lambda=k=1$, $z=0$, in powers of $q=e^{2\pi i \tau}$. The expected result is $$ \chi^1_1=q^{5/24}(2+2q+6q^2+8q^3+\cdots)\tag{14.179} $$

How can I use Mathematica to recover eq.$14.179$ from the other two equations? The naive approach does not quite work, because $\chi$ yields an indeterminate form if we take $\lambda=1,z=0$ directly. And the sum does not converge for $|\mathrm{re}(q)|<1$ (and it cannot be analytically continued), so the expansion around $q=0$ is only asymptotic (not a proper power series in the strict sense).

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Here's the code, if it helps:

Θ[k_, λ_, z_, τ_] := Sum[Exp[-2 π I (k n z + 1/2 λ z - k n^2 τ - n λ τ - λ^2 τ/(4 k))], {n, -∞, ∞}]
χ[k_, λ_, z_, τ_] := (Θ[k + 2, λ + 1, z, τ] - Θ[k + 2, -λ - 1, z, τ])/(Θ[2, 1, z, τ] - Θ[2, -1, z, τ])

Answer 1

Let $p=e^{2\pi iz}$ and $q=e^{2\pi i\tau}$.

thetaTerm[k_, l_] := p^(-k*n - l/2) q^(k*n^2 + l*n + l^2/(4*k))
numTerm = thetaTerm[k+2, l+1] - thetaTerm[k+2, -l-1] /. {l->1, k->1}
denTerm = thetaTerm[2, 1] - thetaTerm[2, -1]
max = 3
numSum = Sum[numTerm/q^(1/3), {n, -max, max}] // Expand
denSum = Sum[denTerm/q^(1/8), {n, -max, max}] // Expand
lim = Limit[numSum/denSum, p->1]
ser = Series[lim, {q, 0, max}]

$2+2 q+6 q^2+8 q^3+O\left(q^4\right)$

The prefactor $q^{5/24}$ comes from the $q^{1/3}$ in the numerator terms and the $q^{1/8}$ in the denominator terms, which I removed.