nested sums

• To: mathgroup at smc.vnet.net
• Subject: [mg125219] nested sums
• From: Severin Pošta <severin at km1.fjfi.cvut.cz>
• Date: Wed, 29 Feb 2012 07:26:19 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

I have noticed that

Sum[x^a y^b z^c, {a, 0, Infinity}, {b, 0, Infinity}, {c, 0, a - b}]

produces "wrong" result

(-y + z - y z + x y z)/((-1 + x) (-1 + y) (y - z) (-1 + x z))

because a-b is not checked to be nonnegative in the nested sums. This is
probably by design (?).
I wonder if I can achieve such check to be perfomed.

This does not work:

Sum[x^a y^b z^c If[a - b >= 0, 1, 0], {a, 0, Infinity}, {b, 0,
Infinity}, {c, 0, a - b}]

This also does not work:

Sum[x^a y^b z^c Piecewise[{{1, a - b >= 0}}], {a, 0, Infinity}, {b, 0,
Infinity}, {c, 0, a - b}]

It is also interesting that Sum[...,{},{}] is not equivalent to
Sum[Sum[...,{}],{}]. The following produces correct result:

s= Sum[x^a y^b z^c Piecewise[{{1, a - b >= 0}}], {c, 0, a - b}];
s1=Sum[s, {b, 0, Infinity}];
Sum[s1,{a, 0, Infinity}]

-(1/((-1 + x) (-1 + x y) (-1 + x z)))

Interesting.

S.

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