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How to simplify this expression?

• To: mathgroup at smc.vnet.net
• Subject: [mg93640] How to simplify this expression?
• From: Barrow <GRseminar at gmail.com>
• Date: Wed, 19 Nov 2008 05:40:37 -0500 (EST)

Dear all,

In my study of physics, I met this expression,

fm = Sum[(-1)^j Binomial[2m,j+m] (StirlingS1[j+m-1,j] + StirlingS1[j
+m-1,j-1]) , {j,1,m}];

This probably can be simplified to (2m-1)!! for m > 1.
However, I tried Simplify or FullSimplify, it didn't works.
But, numerically, I tried m = 2 to m = 400, this identity is correct!
Is there any method to make Mathematica check this identity
symbolically?
Any help will be appreciated. Thanks!

Sincerely
Barrow