Re: Finite Sum

• To: mathgroup at smc.vnet.net
• Subject: [mg44537] Re: Finite Sum
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Thu, 13 Nov 2003 05:57:55 -0500 (EST)
• Organization: The University of Western Australia
• References: <bovefl\$kt3\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <bovefl\$kt3\$1 at smc.vnet.net>,
Ratna Bandila <choppalaks at yahoo.com> wrote:

> Kindly help if anyone has come across the Finite Sum
> expression and has found the closed-form formula for
>
> Sum[(2*(n-m))!/(m!^2*(n-m)!^4),{m,0,n}]

If you enter this expression into Mathematica it automatically returns a
closed-form formula in terms of a (terminating) generalized
hypergeometric function. In general, this will not simplify any further.
If you reverse the summation order, you do get a slightly simpler
result, again in terms of a generalized hypergeometric function:

Sum[(2*(n - m))!/(m!^2*(n - m)!^4) /. m -> n - m, {m, 0, n}]

HypergeometricPFQ[{1/2, -n, -n}, {1, 1}, 4]/Gamma[n + 1]^2

Cheers,
Paul

--
Paul Abbott                                   Phone: +61 8 9380 2734
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Crawley WA 6009                      mailto:paul at physics.uwa.edu.au
AUSTRALIA                            http://physics.uwa.edu.au/~paul

```

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