I am using HypergeometricPFQ functions (more exactly $_3F_2$) as approximants for other more complicated functions. Here are three of them (corresponding to N=3, N=4 and N=5, respectively, in the plots below):
HypergeometricPFQ[{1., -11.359073074960966, -0.9496684005505038}, {-11.842171437570526, -0.6544878051911667}, - \[Tau]]
HypergeometricPFQ[{1., -0.993892734328931, -0.052722890162238745}, {-0.9519055083936611, -0.02621386350907018}, - \[Tau]]
HypergeometricPFQ[{1., -1.0042388354011744, 0.22226474916891914}, {-1.0459210550826585, 0.07845826690277335}, - \[Tau]]
A plot of the functions above in terms of $\tau$ shows that the very first HypergeometricPFQ shown above (the $N=3$ case) goes to a very large negative value when $\tau$ increases while the other ones do not show this type of behavior: plot of HypergeometricPFQ with all cases, the N=3 case decreases much faster than the other ones.
Removing the $N = 3$ case it is possible get a better visualization of the other functions. They all decrease with $\tau$ as well but much slower than the problematic case. Same plot as above, without the N=3 case for better presentation.
I am using the Mathematica software to make these calculations/plots. I am aware that sometimes the default precision is not good to get results with some special functions, but I already took care of that and checked this plot with a large working precision.
At the end of the day I would Laplace transform these functions and end up having Meijer-G functions, which are more complicated and would still show this behavior of diverging to minus infinity as well (I already checked that this is the case for the first HypergeometricPFQ above).
I already read topics and searched on forums
about hypergeometric functions but I failed to understand what
exactly causes the first HypergeometricPFQ to sort of diverge to
minus infinity when $\tau$ increases. My guess is that it has
something to do with large values of the parameters (i.e. -11.842171437570526 and -11.359073074960966) but I was not able to go much further than this.
I already checked that the ratio of the coefficients in the series that defines HypergeometricPFQ go to 1 (because this is a $_pF_q$ where $p=q+1$).
I know $_3F_2(a_1, a_2, a_3; b_1, b_2; -\tau)$ has an infinite number of poles as a function of $b_1$ and $b_2$. My understanding was that after fixing $b_1$ and $b_2$ to a value that is not a pole, the function would have a smooth behavior with respect to $\tau$, which is not the case for $N=3$. So, I do not know exactly what causes this behavior in the $N=3$ case.
I really appreciate any help, references, etc...
For hypergeometric functions $_NF_{N-1}$, there exist connection formulas relating their behavior at $0$ and $\infty$, such as these: 1, 2 (for generic parameters).
Looking at the asymptotics of the r.h.s. as $z\to-\infty$, you may see that $_NF_{N-1}\left(\begin{array}{c} a_1,\ldots a_N \\ b_1,\ldots b_{N-1}\end{array};z\right)$ behaves, up to an explicitly known constant prefactor, as $(-z)^{\max_{k}\{-a_k\}}$. Thus it is the value $a_2=-11.35907$ which is responsible for faster divergence. Does this answer your question?