paratusojitos0yx

2022-11-17

Doing some computations and plottings I've found out that the function
${}_{2}{F}_{1}\left(2s-1,s-\frac{1}{2};s;-1\right)$
behaves for large real s like ${4}^{-s}$. More precisely: It seems that
${}_{2}{F}_{1}\left(2s-1,s-\frac{1}{2};s;-1\right){4}^{s}$
grows, but very very slowly. No exponential growth or decay at all. If you modify the 4 only slightly this of course changes and you get exponential groth or decay. So my question is to explain this phenomenon. Is

Is there an easy to detemine for every x>0 the respective b(x)>0 such that you have
${}_{2}{F}_{1}\left(2s-1,s-\frac{1}{2};s;-x\right)\sim {b}^{-s}?$
What is

then?

siriceboynu1

$f\left(s\right)={\phantom{\rule{thinmathspace}{0ex}}}_{2}{F}_{1}\left(2s-1,s-\frac{1}{2};s;-1\right)={\phantom{\rule{thinmathspace}{0ex}}}_{2}{F}_{1}\left(s-\frac{1}{2},2s-1;s;-1\right)$
Even for small values of s, we have a nice logarithmic behaviour
$\mathrm{log}\left[f\left(s\right)\right]\sim a-b\phantom{\rule{thinmathspace}{0ex}}s$
Using the data computed for $1\le n\le 100$, we have, with ${R}^{2}=0.9999982$,
$\begin{array}{clcl}& \text{Estimate}& \text{Standard Error}& \text{Confidence Interval}\\ a& 1.640373& 0.021344& \left\{1.598011,1.682736\right\}\\ b& 1.379308& 0.000367& \left\{1.380037,1.378580\right\}\end{array}$
and, as you noticed, b is quite close to $\mathrm{log}\left(4\right)=1.38629$
Pushing the numerical analysis much further, b is closer and closer to $\left(\mathrm{log}\left(4\right)-ϵ\right)$. This is normal since
${\phantom{\rule{thinmathspace}{0ex}}}_{2}{F}_{1}\left(2s,s;s;-1\right)={4}^{-s}$
What is interesting is that, if $\mathrm{log}\left[f\left(s\right)\right]$ is an increasing function going to infinity while, if $\mathrm{log}\left[f\left(s\right)\right]$ goes through a maximum value.
What is interesting is that
${4}^{s}{\phantom{\rule{thinmathspace}{0ex}}}_{2}{F}_{1}\left(s-1,2s-1;s;-1\right)=2+2\sqrt{\pi }\frac{\mathrm{\Gamma }\left(s\right)}{\mathrm{\Gamma }\left(s-\frac{1}{2}\right)}$

Do you have a similar question?