# Should [tag:hypergeometric-function] be generalized to admit higher weights, or should a new tag be created?

Right now, the tag wiki and excerpt of refer specifically to the Gaussian hypergeometric function $$_2F_1(a,b;c;z)$$, functions of the form $$_2F_1(a,b;c;z) = \sum_{n=0}^{\infty}\frac{(a)_n (b)_n}{(c)_n n!}z^n,$$where $$(a)_n= a\cdot (a+1)\cdot\ldots \cdot (a+n-1)$$ is the Pochhammer symbol. This function is ubiquitous and important: the Maclaurin series of many elementary functions can be expressed this way, and the $$_2F_1$$ function is central in the theory of ODEs. It is a very active and interesting tag on the main site showcasing creative and technical work from many users and I hope it will remain so.

However, the $$_2F_1$$ function is a special case in a family of similar functions $$_p F_q$$ that admit $$p$$ and $$q$$ Pochhammer parameters in the numerator/denominator, respectively. These functions appear naturally in the study of log integrals and multiple zeta functions, or MZVs; examples can be found here, here, and here. The problem is that, strictly speaking, their usage is not covered by the current tag. This is relevant because identities and simplifications for these functions depend heavily on the number of parameters, and may not extend from $$_2F_1$$ to the general case.

I propose two options to rectify this problem:

1. Expand the excerpt and wiki of the current tag to include the general case.
2. Create a new tag or possibly rename the current one to emphasize it is a (valid) special case.
• I agree on including ${}_pF_q$ in the existing tag. Indeed, some questions have already used it that way. Oct 16, 2020 at 11:25
• People will use it that way unless there's a more specific tag, and possibly even if there is. (But I also don't think there's enough $_pF_q$ questions to justify a separate tag, and it's better to edit the current one's description.) Oct 16, 2020 at 18:19
• Option 1 seems to be a good middle course. The hypergeometric functions have extensive generalizations, but merely including ${}_pF_q$ is a natural limitation. Oct 17, 2020 at 14:12
• @hardmath I only mentioned the $_pF_q$ case as that is what I'm most familiar with (and then only superficially). I will wait a few more days but it seems the general sentiment is 'expand/update the current tag'. Oct 17, 2020 at 14:37

I have edited the tag wiki to include the $$_p F _q$$ case and gave a mention to several extensions, as hardmath suggested. Please feel free to improve the blurb.