# Another example where using FullSimplify gives different result than Simplify

I believe this question is very similar to Result of Series[expression] is different when I simplify the expression, however, due to my lack of Mathematica experience, I am reluctant to call it a bug. The issue is that when taking a series expansion of a simplified expression, it yields a different result than the series of the same expression, but fully simplified. The original expression I want to series expand is quite long, and it is defined below:

expression[ω_] := (12 I E^(-I π ω -
I π (2 + ω)) (12 E^(I π ω)
Gamma[5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] -
12 E^(3 I π ω)
Gamma[5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] -
22 E^(I π ω) ω Gamma[
5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] +
22 E^(3 I π ω) ω Gamma[
5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] +
36 E^(I π ω) ω^2 Gamma[
5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] -
36 E^(3 I π ω) ω^2 Gamma[
5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] +
26 E^(I π ω) ω^3 Gamma[
5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] -
26 E^(3 I π ω) ω^3 Gamma[
5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] -
36 E^(I π ω) ω^4 Gamma[
5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] +
36 E^(3 I π ω) ω^4 Gamma[
5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] +
8 E^(I π ω) ω^5 Gamma[
5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] -
8 E^(3 I π ω) ω^5 Gamma[
5 - 2 ω] HypergeometricPFQ[{5 - 2 ω,
4 - ω, -ω}, {3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] +
3 E^(I π ω)
Gamma[5 - 2 ω] HypergeometricPFQ[{2, 5 - 2 ω,
4 - ω, -ω}, {1, 3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] -
3 E^(3 I π ω)
Gamma[5 - 2 ω] HypergeometricPFQ[{2, 5 - 2 ω,
4 - ω, -ω}, {1, 3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] -
4 E^(I π ω) ω Gamma[
5 - 2 ω] HypergeometricPFQ[{2, 5 - 2 ω,
4 - ω, -ω}, {1, 3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] +
4 E^(
3 I π ω) ω Gamma[
5 - 2 ω] HypergeometricPFQ[{2, 5 - 2 ω,
4 - ω, -ω}, {1, 3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] +
7 E^(I π ω) ω^2 Gamma[
5 - 2 ω] HypergeometricPFQ[{2, 5 - 2 ω,
4 - ω, -ω}, {1, 3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] -
7 E^(3 I π ω) ω^2 Gamma[
5 - 2 ω] HypergeometricPFQ[{2, 5 - 2 ω,
4 - ω, -ω}, {1, 3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] +
10 E^(I π ω) ω^3 Gamma[
5 - 2 ω] HypergeometricPFQ[{2, 5 - 2 ω,
4 - ω, -ω}, {1, 3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] -
10 E^(3 I π ω) ω^3 Gamma[
5 - 2 ω] HypergeometricPFQ[{2, 5 - 2 ω,
4 - ω, -ω}, {1, 3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] -
4 E^(I π ω) ω^4 Gamma[
5 - 2 ω] HypergeometricPFQ[{2, 5 - 2 ω,
4 - ω, -ω}, {1, 3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] +
4 E^(3 I π ω) ω^4 Gamma[
5 - 2 ω] HypergeometricPFQ[{2, 5 - 2 ω,
4 - ω, -ω}, {1, 3 - ω, 7 - ω},
1] Pochhammer[1 + 1/2 (4 - 2 ω),
1 + ω] Pochhammer[2 - ω, 1 + ω] +
240 E^(I π (2 + ω))
Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] -
240 E^(2 I π ω + I π (2 + ω))
Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] -
232 E^(I π (2 + ω)) ω Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] +
232 E^(2 I π ω +
I π (2 + ω)) ω Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] -
40 E^(I π (2 + ω)) ω^2 Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] +
40 E^(2 I π ω +
I π (2 + ω)) ω^2 Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] +
282 E^(I π (2 + ω)) ω^3 Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] -
282 E^(2 I π ω +
I π (2 + ω)) ω^3 Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] -
201 E^(I π (2 + ω)) ω^4 Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] +
201 E^(2 I π ω +
I π (2 + ω)) ω^4 Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] +
50 E^(I π (2 + ω)) ω^5 Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] -
50 E^(2 I π ω +
I π (2 + ω)) ω^5 Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] -
4 E^(I π (2 + ω)) ω^6 Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω, 1 + ω] +
4 E^(2 I π ω +
I π (2 + ω)) ω^6 Gamma[-2 (-2 + ω)] Pochhammer[5 - 2 ω,
1 + ω] Pochhammer[-2 +
1/2 (4 - 2 ω) + ω,
1 + ω]))/((-6 + ω) (-5 + ω) (-4 + ω) (-3 + ω) (-2 + ω) (-1 + ω) ω (1 + ω) Gamma[-2 (-2 + ω)] Pochhammer[
1 + 1/2 (4 - 2 ω), 1 + ω] Pochhammer[2 - ω,
1 + ω]);


The command

expression[ω] // Simplify;
SeriesCoefficient[%, {ω, 0, 0}]


outputs

2π

while the command

expression[ω] // FullSimplify;
SeriesCoefficient[%, {ω, 0, 0}]


outputs

0

Can anyone tell me if it is a bug or am I missing something?

The complicated expression ("expression[ω]") is an putput of the following command line

ElementSum[n_,\[Omega]_]:= -((12 I E^(-I \[Pi] (2 n + \[Omega])) (E^(2 I n \[Pi]) - E^(
2 I \[Pi] \[Omega])) (-20 + 6 \[Omega] + 9 \[Omega]^2 -
18 \[Omega]^3 + 4 \[Omega]^4 +
n^2 (1 - 2 \[Omega] + 4 \[Omega]^2) +
n (4 - 10 \[Omega] + 20 \[Omega]^2 - 8 \[Omega]^3)) Gamma[
4 + n - 2 \[Omega]])/((-1 + n - \[Omega]) (n - \[Omega]) (1 +
n - \[Omega]) (3 + n - \[Omega]) (4 + n - \[Omega]) (5 +
n - \[Omega]) n! Gamma[4 - 2 \[Omega]]));

expression[ω_]:=  Series[ElementSum[n,\[Omega]],{n,0,Infinity}]

• It does seem to be peculiar behavior. – Daniel Lichtblau Mar 8 at 18:14
• The expression and its simplified forms are undefined at omega == 0; although the FullSimplify form exists in the Limit (limit is zero as expected from its Series). – Bob Hanlon Mar 8 at 18:50
• Thank you for your comment Mr.Hanlon. Correct me if I'm wrong, but I thougth that the limit of the raw expression, its simplified and fullysimplified versions should agree at any given point, because they would be symbolically equivalent, it does not seem to be the case here. – Gabriel Nagaoka Mar 8 at 20:20
• Where did this long series of hypergeometrics come from? Looks to me that this would have been better off reformulated in terms of Meijer $G$. – J. M. will be back soon Mar 8 at 23:50
• Thanks for the comment J. M., this series of hypergeometrics comes from the infinite sum I will post as an edit to the original question. Could you elaborate in the Meijer G if possible? – Gabriel Nagaoka Mar 10 at 0:47