Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I would be grateful if anyone could explain the purpose of using hypergeometric functions. If a function exists in closed form, e.g. $\sum\limits_{k \geq 0}z^k = {}_2 F_1 \bigg[{{1\; 1}\atop{1}} \vert z \bigg] = \frac{1}{1-z}$, but in case it doesn't, why bother rewriting it, e.g. $\sum\limits_{k \leq m}\binom{n}{k} = \sum\limits_{k \geq 0} \binom{n}{m-k} = \binom{n}{m} {}_2 F_1 \bigg[{{-m\; 1}\atop{n-m+1}} \vert 1 \bigg] $

since it doesn't yield a closed form or approximation of it?

share|improve this question
Closed form is in the eye of the beholder. –  André Nicolas Feb 8 '12 at 22:12
Let's take your second sum as an example; it turns out that knowing the $-m$ numerator parameter is a negative integer is crucial, since it is a known property of hypergeometric functions that they degenerate to polynomials whenever one or both of their numerator parameters are negative. –  Guess who it is. Feb 8 '12 at 22:37

1 Answer 1

up vote 2 down vote accepted

Hmm, I don't know... because the Gaussian hypergeometric function satisfies a very convenient set of identities?

Also, what André said in the comments. Gauss and others spent a fair bit of time unraveling identities satisfied by this function, and it'd be a damn shame not to make use of our predecessors' effort.

share|improve this answer
Re: approximation, you honestly don't think that all hypergeometric functions are always evaluated through the series, do you? –  Guess who it is. Feb 8 '12 at 22:27
All I'm thinking is that hypergeometric transformation might help in finding asymptotic lower and upper bounds on the sum –  user19821 Feb 9 '12 at 0:50
Right; since the asymptotic behavior of hypergeometric functions are well-studied (did you have a look at the links I gave?), among other things, being able to represent something in terms of a hypergeometric function is a great boon. Hypergeometric functions also satisfy well-studied difference and differential equations, and those viewpoints are also a gold mine for teasing out useful properties. –  Guess who it is. Feb 9 '12 at 0:58
Thanks. Can you recommend some articles where they are applied in this way? –  user19821 Feb 9 '12 at 4:39
If you look through the DLMF, the bibliography there should get you started... –  Guess who it is. Feb 9 '12 at 4:44

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.