# Series expansion error in a hypergeometric function

Bug introduced in 5.0 or earlier and fixed in 11.2

Compared to the Maple 2016 result of the series expansion around $x=0$ of ${}_2F_1(1,1;-1/3;1-x)$ up to order 3, Mathematica 11.1 seems to return an incorrect result. I may be wrong so could someone please check the details?

The thing is both could be wrong, although below I compared them to the exact evaluation, and the Maple solution is closer to the result of an exact computation.

According to the Maple 2016 online document

Looks like the marked term is missing in the Mathematica solution, as presented below:

f[x_] = Normal[FullSimplify[Series[Hypergeometric2F1[1, 1, -(1/3), 1 - x], {x, 0, 3}]]]


Comparing Mathematica (f, blue) and Maple (g, yellow), Maple gives better approx. so it could be correct:

g[x_] := f[x] - 64/6561 π Sqrt[3] x^(8/3)
errf[x_] := f[x] - Hypergeometric2F1[1, 1, -1/3, 1 - x];
errg[x_] := g[x] - Hypergeometric2F1[1, 1, -1/3, 1 - x];
Plot[{errf[x], errg[x]}, {x, 0, 1}, Axes -> False, Frame -> True]


This is not a proof, but just an indication. Could someone please look at the details and "formally" conclude if just one of them wrong, or both, and what is going on?

• This seems more like an answer than a question. You should report errors/ bugs to Wolfram. (Menu Help > Give Feedback...) May 13, 2017 at 22:10
• People here generally like users to post code as Mathematica code instead of images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find this this meta Q&A helpful May 13, 2017 at 22:10
• Reported as a bug. May 16, 2017 at 15:23
• Looks like this one has been around since version 5.2. May 17, 2017 at 10:48

It's present for me:

Of course, this one is missing the 11/3 power. I'm not sure why the term shows up only if the requested order is more than one higher than 8/3. A bit counterintuitive, admittedly.

Update: In general, it appears that the series for

s = Series[Hypergeometric2F1[1, 1, a/b, 1 - x], {x, 0, n}]


where $a$ and $b$ are integers and $a/b$ is not an integer, has the form

$$s = p_n(x) + x^{-2+{a \over b}}\,q_{n+1}(x)$$

where $p_n$ is a degree $n$ polynomial and $q_{n+1}$ is a polynomial of up to degree $n+1$, where the degree is truncated at either $n+1$ or $n+1+{a \over b}$, whichever is less.

This could be better in some cases, except the big-$O$ term is incorrect:

Series[
Series[Hypergeometric2F1[1, 1, a/b, 1 - x], {x, 0,Ceiling@Max@{n, n - a/b + 1}}],
{x, 0, n}]


For the OP's a/b = -1/3 and n = 3:

• Seems also to happen for Hypergeometric2F1[1, 1, -1/n, 1 - x]. May 13, 2017 at 22:24
• This is the error, $x^{8/3}$ should be presented if the order limit is set to $3$, the same for $11/3$ in a case of $n=4$. May 13, 2017 at 22:27
• @ahrvoje Yes, just so. Please report it to WRI. May 13, 2017 at 22:32

As a workaround, use the Pfaff transformation:

Series[x^(-7/3) Hypergeometric2F1[-4/3, -4/3, -1/3, 1 - x], {x, 0, 3}]


which has the -((64 π x^(8/3))/(2187 Sqrt[3])) term that is missing from Series[Hypergeometric2F1[1, 1, -1/3, 1 - x], {x, 0, 3}].