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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?

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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. –  J. M. Feb 8 '12 at 22:37

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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.

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Re: approximation, you honestly don't think that all hypergeometric functions are always evaluated through the series, do you? –  J. M. 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. –  J. M. 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... –  J. M. Feb 9 '12 at 4:44

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