# Understanding the behavior of HypergeometricPFQ

this is my very post here, so I apologize for any possible format issue.

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:

$\tau$. " />

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. 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.

$\tau$, without the N=3 case." />

At the end of the day I would take the Laplace Transform of 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 (here on the forum and other books) about the hypergeometric functions but I failed to understand what exactly causes the first HypergeometricPFQ to sort of diverge to minus infinity. My guess is that it has something to do with large values of the parameters (i.e. -11.842171437570526 and -11.359073074960966) because this is the 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 $$p=q+1$$), so I do not know what causes this sort of behavior in what I called $$N=3$$ case.

I really appreciate any help. References are welcome.

EDIT:

The behavior of these hypergeometric functions at large $$\tau$$ can be understood via this expression, which show they behave like $$\tau^{-max(a_k)}$$ where $$a_k$$ is a parameter figuring in $$_{q+1}F_q({\bf a},{\bf b},\tau)$$ (I am using boldface variables as vectors: $${\bf a} = (a_1, ..., a_m)$$).

Thank you for everyone that took time to write here!

• This question appears to be about the math rather than Mathematica. Perhaps you should ask at Mathematics StackExchange – Bob Hanlon Apr 22 at 3:24
• @Bob Hanlon I think explaining the best Mathematica tools to understand the behaviour of a family of functions is within the scope of this site. – mikado Apr 22 at 5:44
• The parameters {1., -11.359073074960966, -0.9496684005505038}, {-11.842171437570526, -0.6544878051911667} of the first one substantially differ from the parameters {1., -0.993892734328931, -0.052722890162238745}, {-0.9519055083936611, -0.02621386350907018} of the second one. This implies the difference of their behaviors for big positive values of \[Tau]. The question should be closed as meaningless. – user64494 Apr 22 at 9:16
• -1. I wonder the upvotes. – user64494 Apr 22 at 9:28
• @BobHanlon thank you the suggestion. I will try posting there. – avg Apr 22 at 12:14