I would like to expand the following function at $x \sim 0$ up to some high $x_\text{max} = \Delta_\text{max}$:
$$16 \sum_{\Delta=1}^{\Delta_\text{max}} \sum_{s=0}^{\Delta-2} f_{\Delta,s} \frac{(s+\Delta)(1+s-\Delta)}{(s-\Delta)(1+s+\Delta)}\ _2F_1 \left(\frac{1}{2},-s,\frac{1}{2}-s,1\right)\ _2F_1 \left(\frac{1}{2},2+\Delta,\frac{5}{2}+\Delta,x^2 \right) x^{2+\Delta}, \tag{1}$$
where $f_{\Delta,s}$ are numerical coefficients. The series expansion should be performed after doing the following replacement (everywhere except in the subscript of $f_{\Delta,s}$ and in the summation limits):
$$\Delta \to \Delta + g^2 \Delta^{(2)}_{\Delta,s} + g^4 \Delta^{(4)}_{\Delta,s}, \tag{2}$$
and expanding in $g$. Again the $\Delta^{(k)}$ are numbers, which depend on $\Delta$ and $s$. I am not interested in $\log$ terms, which can be discarded by setting the rule $\log x \to 0$.
So far I wrote the following simple code:
Δmax=5;
ℒ0s2 = 1/((s - Δ) (1 + s + Δ)) 16 x^(2 + Δ) (1 + s - Δ) (s + Δ) Hypergeometric2F1[1/2, -s, 1/2 - s, 1] Hypergeometric2F1[1/2, 2 + Δ, 5/2 + Δ, x^2]
Sum[Series[(f[Δ, s] Series[ℒ0s2 /. {Δ -> \
Δ + g^2 Δ2[Δ, s] + g^4 Δ4[Δ, s]}, {g, 0, 4}] /. {Log[x] -> 0} // Normal) /. {Δ -> ΔΔ, s -> ss}, {x, 0, Δmax}] /. _?(N[#] == 0 &) :> 0 // Normal, {ΔΔ, 2, Δmax}, {ss, 0, ΔΔ - 2}]
Unfortunately this is insanely inefficient for high $\Delta_\text{max}$. For $\Delta_\text{max}=5$ the computation takes less than a second, for $\Delta_\text{max}=10$ about $20$ seconds, for $\Delta_\text{max}=15$ it took $3$ minutes,... I let it run the whole night yesterday to reach my goal, which is at least $\Delta_\text{max}=60$, and it was still running this morning!
Any suggestion how this expansion could be done (much) faster?
