Polynomial CoeÆients
William J. Wedemeyer, Harold A. Sheraga
Thisrstsupplementgivestheseventeen looppolynomialoeÆientsR
0
-R
16
[orresponding to equation (7) of the text℄ as analyti formulae in the
known quantities P
00 P
22
and Q
00 Q
44
. These lengthy formulae were
generated using the software pakage Maple and are inluded for
omplete-ness. However, the oeÆients R
0 -R
16
an be alulated more eÆiently by
The R
0
CoeÆient 3
The R
1
CoeÆient 4
The R
2
CoeÆient 5
The R
3
CoeÆient 6
The R
4
CoeÆient 8
The R
5
CoeÆient 12
The R
6
CoeÆient 17
The R
7
CoeÆient 24
The R
8
CoeÆient 31
The R
9
CoeÆient 39
The R
10
CoeÆient 46
The R
11
CoeÆient 53
The R
12
CoeÆient 58
The R
13
CoeÆient 62
The R
14
CoeÆient 64
The R
15
CoeÆient 65
The R
16
0
R
0
= Q
00 Q
04 P
4
01 +P
2
00 Q
2
02 P
2
02 +P
4
00 Q
2
04
2Q
02 P
3
00 Q
04 P
02 +P
2
00 Q
02 Q
04 P
2
01 +Q
2
00 P
4
02 +
Q
00 P
2
01 Q
02 P
2
02 4Q
00 P
2
01 P
00 Q
04 P
02 +2Q
00 P
2
00 P
2
02 Q
04
2Q
00 P
00 P
3
02 Q
1
R
1
= P
00 Q
11 Q
04 P
3
01 +4P
3
00 P
10 Q
2
04 4Q
00 P
2
01 P
00 Q
04 P
12
4Q
00 P
2
01 P
10 Q
04 P
02 6Q
00 P
00 P
2
02 Q
02 P
12 +4Q
00 P
2
00 P
02 Q
04 P
12 +
4Q
00 P
10 P
2
02 P
00 Q
04 6P
2
00 Q
02 P
10 Q
04 P
02 2Q
02 P
3
00 Q
04 P
12 +
2P 2
00 Q
2
02 P
02 P
12 Q
00 P
3
01 Q
13 P
02 +4Q
00 P
3
01 Q
04 P
11 +
2P
00 P
10 Q
2
02 P
2
02 +2Q
00 P
2
01 Q
02 P
02 P
12 2Q
00 P
10 P
3
02 Q
02
Q
00 P
01 Q
11 P
3
02 +3Q
00 P
01 P
00 P
2
02 Q
13 8Q
00 P
01 P
11 P
00 Q
04 P
02 +
4Q 2
00 P
3
02 P
12 +2P
00 P
10 Q
02 Q
04 P
2
01 P
00 Q
11 P
01 Q
02 P
2
02 +
3Q
11 P
01 P
2
00 Q
04 P
02 P
2
00 Q
02 P
01 Q
13 P
02 +2P
2
00 Q
02 P
01 Q
04 P
11
P 3
00 Q
13 Q
04 P
01 +2Q
00 P
01 P
11 Q
02 P
2
2
R
2
= 2P 4
00 Q
04 Q
24 +2Q
00 Q
20 P
4
02 +Q
20 Q
04 P
4
01 +
P 2
00 Q
2
02 P
2
12 +6Q
2
00 P
2
02 P
2
12 +4P
3
00 P
20 Q
2
04
2Q
00 P
00 P
3
02 Q
22 6Q
00 P
00 P
2
02 Q
02 P
22 +4Q
00 P
2
00 P
02 Q
04 P
22 +
2Q
00 P
2
00 P
2
12 Q
04 6Q
00 P
10 P
2
02 Q
02 P
12 +8Q
00 P
10 P
02 P
00 Q
04 P
12 +
2Q
00 P
2
00 P
2
02 Q
24 +4Q
00 P
00 P
2
02 P
20 Q
04 +2Q
00 P
2
10 P
2
02 Q
04
2Q
00 P
20 P
3
02 Q
02 6P
00 P
2
10 Q
02 Q
04 P
02 2P
00 Q
11 P
01 Q
02 P
02 P
12 +
Q
20 P
2
01 Q
02 P
2
02 4Q
20 P
2
01 P
00 Q
04 P
02 P
00 Q
11 P
11 Q
02 P
2
02
2Q
20 P
00 P
3
02 Q
02 2Q
11 P
2
00 P
2
02 Q
13 +P
00 Q
11 P
2
01 Q
13 P
02 +
6P
00 Q
11 P
01 P
10 Q
04 P
02 3P
00 Q
11 P
2
01 Q
04 P
11 +2P
2
00 Q
02 Q
22 P
2
02 +
P 2
00 Q
02 Q
24 P
2
01 +P
3
00 Q
2
13 P
02 +3Q
11 P
01 P
2
00 Q
04 P
12 +
3Q
11 P
11 P
2
00 Q
04 P
02 2Q
02 P
3
00 Q
24 P
02 P
3
00 Q
13 Q
04 P
11
2Q
22 P
3
00 Q
04 P
02 +P
2
00 Q
02 P
2
11 Q
04 P
2
00 Q
02 P
11 Q
13 P
02
P 2
00 Q
02 P
01 Q
13 P
12 +2P
2
00 Q
02 P
01 Q
04 P
21 +2P
2
00 Q
2
02 P
02 P
22 +
P 2
00 Q
22 Q
04 P
2
01 3P
2
00 P
10 Q
13 Q
04 P
01 6P
2
00 Q
02 P
10 Q
04 P
12 +
2P
00 P
20 Q
2
02 P
2
02 2Q
02 P
3
00 Q
04 P
22 6P
2
00 Q
02 P
20 Q
04 P
02 +
4P
00 P
10 Q
2
02 P
02 P
12 2P
00 P
10 Q
02 P
01 Q
13 P
02 +4P
00 P
10 Q
02 P
01 Q
04 P
11 +
P 2
10 Q
2
02 P
2
02 +4Q
2
00 P
3
02 P
22 +6P
2
00 P
2
10 Q
2
04 +
Q
00 Q
24 P
4
01 +P
00 Q
2
11 P
3
02 +2Q
20 P
2
00 P
2
02 Q
04
Q
00 P
3
01 Q
13 P
12 +4Q
00 P
3
01 Q
04 P
21 3Q
00 P
01 Q
11 P
2
02 P
12 +
6Q
00 P
01 P
00 P
02 Q
13 P
12 +4Q
00 P
01 P
11 Q
02 P
02 P
12 P
10 Q
11 P
01 Q
02 P
2
02
P
10 Q
11 Q
04 P
3
01 +P
2
10 Q
02 Q
04 P
2
01 +2P
00 P
20 Q
02 Q
04 P
2
01
4Q
00 P
2
01 P
20 Q
04 P
02 +3Q
00 P
01 P
10 P
2
02 Q
13 Q
00 P
11 Q
11 P
3
02 +
3Q
00 P
11 P
00 P
2
02 Q
13 4Q
00 P
2
11 P
00 Q
04 P
02 +Q
00 P
2
11 Q
02 P
2
02
6Q
00 P
00 P
02 Q
02 P
2
12 8Q
00 P
01 P
11 P
00 Q
04 P
12 8Q
00 P
01 P
11 P
10 Q
04 P
02
8Q
00 P
01 P
21 P
00 Q
04 P
02 +2Q
00 P
01 P
21 Q
02 P
2
02 +Q
00 P
2
01 Q
22 P
2
02 +
Q
00 P
2
01 Q
02 P
2
12 +6Q
00 P
2
01 P
2
11 Q
04 +2Q
00 P
2
01 Q
02 P
02 P
22
4Q
00 P
2
01 P
00 Q
04 P
22 3Q
00 P
2
01 P
11 Q
13 P
02 4Q
00 P
2
01 P
10 Q
04 P
12
4Q
00 P
2
01 P
00 Q
24 P
3
R
3
= 4Q 2
00 P
02 P
3
12 P
20 Q
11 P
01 Q
02 P
2
02 P
20 Q
11 Q
04 P
3
01 +
2P 2
10 Q
2
02 P
02 P
12 P
2
10 Q
02 P
01 Q
13 P
02 +2P
2
10 Q
02 P
01 Q
04 P
11 +
2P
10 P
20 Q
02 Q
04 P
2
01 +2P
10 P
20 Q
2
02 P
2
02 2P
3
10 Q
02 Q
04 P
02
6P 2
00 Q
22 P
10 Q
04 P
02 P
2
00 Q
22 P
01 Q
13 P
02 +2P
2
00 Q
22 P
01 Q
04 P
11
2Q
22 P
3
00 Q
04 P
12 +2P
2
00 Q
2
02 P
12 P
22 +3Q
20 P
01 P
00 P
2
02 Q
13
4Q
20 P
2
01 P
10 Q
04 P
02 3P
2
00 P
10 Q
13 Q
04 P
11
+12P 2
00 P
10 P
20 Q
2
04
3Q
00 P
01 P
2
11 Q
13 P
02 8Q
00 P
01 P
11 P
10 Q
04 P
12 8Q
00 P
01 P
11 P
00 Q
24 P
02
8Q
00 P
01 P
11 P
20 Q
04 P
02 +4Q
00 P
01 P
21 Q
02 P
02 P
12 +2Q
00 P
2
01 Q
22 P
02 P
12 +
2Q
00 P
2
01 Q
02 P
12 P
22 3Q
00 P
2
01 P
11 Q
13 P
12
+12Q
00 P
2
01 P
11 Q
04 P
21
3Q
00 P
2
01 P
21 Q
13 P
02 4Q
00 P
2
01 P
00 Q
24 P
12 3P
2
00 P
20 Q
13 Q
04 P
01 +
8P 3
00 P
10 Q
04 Q
24 +2P
2
00 Q
02 P
01 Q
24 P
11 +P
10 Q
2
11 P
3
02
P 2
00 Q
02 P
01 Q
33 P
02 2Q
02 P
3
00 Q
24 P
12 P
3
00 Q
13 Q
04 P
21
P 3
00 Q
33 Q
04 P
01 +4P
00 P
20 Q
2
02 P
02 P
12 8Q
00 P
11 P
21 P
00 Q
04 P
02 +
2Q
00 P
11 P
21 Q
02 P
2
02 +3Q
00 P
21 P
00 P
2
02 Q
13 +3Q
00 P
11 P
10 P
2
02 Q
13
6Q
00 P
00 P
2
02 Q
22 P
12
12Q
00 P
00 P
02 Q
02 P
12 P
22 2Q
00 P
00 P
3
12 Q
02 +
4Q
00 P
2
00 P
12 Q
04 P
22 2Q
00 P
10 P
3
02 Q
22 6Q
00 P
10 P
02 Q
02 P
2
12
6Q
00 P
10 P
2
02 Q
02 P
22 +8Q
00 P
10 P
02 P
00 Q
04 P
22 +4Q
00 P
2
00 P
02 Q
24 P
12 +
8Q
00 P
00 P
02 P
20 Q
04 P
12 +4Q
00 P
00 P
2
02 P
10 Q
24 +3Q
00 P
01 P
00 P
2
02 Q
33 +
2Q
00 P
01 P
11 Q
22 P
2
02 +4Q
00 P
01 P
11 Q
02 P
02 P
22 8Q
00 P
01 P
11 P
00 Q
04 P
22 +
4P
00 P
3
10 Q
2
04 +P
3
00 Q
2
13 P
12 P
3
00 Q
13 Q
24 P
01 +
6P
00 Q
11 P
01 P
20 Q
04 P
02 2P
00 Q
11 P
01 Q
02 P
02 P
22 4Q
00 P
2
01 P
10 Q
04 P
22
4Q
00 P
2
01 P
20 Q
04 P
12 4Q
00 P
2
01 P
10 Q
24 P
02 8Q
00 P
01 P
21 P
00 Q
04 P
12
8Q
00 P
01 P
21 P
10 Q
04 P
02 +3Q
00 P
01 P
00 P
2
12 Q
13 +6Q
00 P
01 P
10 P
02 Q
13 P
12 +
3Q
00 P
01 P
20 P
2
02 Q
13 3Q
00 P
11 Q
11 P
2
02 P
12 +6Q
00 P
11 P
00 P
02 Q
13 P
12 +
2Q
00 P
2
11 Q
02 P
02 P
12 4Q
00 P
2
11 P
00 Q
04 P
12 4Q
00 P
2
11 P
10 Q
04 P
02
2P
00 P
10 Q
02 P
01 Q
13 P
12 +4P
00 P
10 Q
02 P
01 Q
04 P
21
+12Q 2
00 P
2
02 P
12 P
22 +
4P
00 P
10 Q
02 Q
22 P
2
02 +2P
00 P
10 Q
02 Q
24 P
2
01 +2P
00 P
10 Q
02 P
2
11 Q
04
2P
00 P
10 Q
02 P
11 Q
13 P
02 +8Q
00 Q
20 P
3
02 P
12 Q
00 P
3
01 Q
13 P
22
Q
00 P
3
01 Q
33 P
02 +4Q
00 P
3
01 Q
24 P
11 Q
00 P
01 Q
31 P
3
4Q
00 P
01 P
11 Q
04 3Q
00 P
01 Q
11 P
02 P
22 3Q
00 P
01 Q
11 P
02 P
12 +
6Q
00 P
01 P
00 P
02 Q
13 P
22 Q
00 P
21 Q
11 P
3
02 +4Q
00 P
00 P
2
12 P
10 Q
04 +
4Q
00 P
2
10 P
02 Q
04 P
12 +4Q
00 P
10 P
2
02 P
20 Q
04 6Q
00 P
20 P
2
02 Q
02 P
12
2P
00 P
20 Q
02 P
01 Q
13 P
02 +4P
00 P
20 Q
02 P
01 Q
04 P
11 6P
00 P
2
10 Q
02 Q
04 P
12
12P
00 P
10 Q
02 P
20 Q
04 P
02 3P
00 P
2
10 Q
13 Q
04 P
01 +3Q
11 P
11 P
2
00 Q
04 P
12 +
P
10 Q
11 P
2
01 Q
13 P
02 3P
10 Q
11 P
2
01 Q
04 P
11 P
10 Q
11 P
11 Q
02 P
2
02
2P
10 Q
11 P
01 Q
02 P
02 P
12 4P
00 Q
11 P
10 P
2
02 Q
13 P
00 Q
31 P
01 Q
02 P
2
02 +
4P 2
00 Q
02 Q
22 P
02 P
12 P
2
00 Q
02 P
11 Q
13 P
12 +2P
2
00 Q
02 P
11 Q
04 P
21
P 2
00 Q
02 P
21 Q
13 P
02 6P
2
00 Q
02 P
10 Q
04 P
22 6P
2
00 Q
02 P
20 Q
04 P
12
6P 2
00 Q
02 P
10 Q
24 P
02 +3P
2
00 P
10 Q
2
13 P
02 +2P
00 P
10 Q
2
02 P
2
12 +
4P
00 P
10 Q
2
02 P
02 P
22 +2P
00 P
10 Q
22 Q
04 P
2
01 P
2
00 Q
02 P
01 Q
13 P
22 +
3P 2
10 Q
11 P
01 Q
04 P
02 +2Q
20 P
01 P
11 Q
02 P
2
02 +4Q
20 P
3
01 Q
04 P
11 +
6P
00 Q
11 P
01 P
10 Q
04 P
12 +P
00 Q
11 P
2
01 Q
13 P
12 +4Q
20 P
2
00 P
02 Q
04 P
12
4Q
11 P
2
00 P
02 Q
13 P
12 6Q
20 P
00 P
2
02 Q
02 P
12 2Q
20 P
10 P
3
02 Q
02
P
00 Q
11 P
01 Q
22 P
2
02 P
00 Q
11 P
01 Q
02 P
2
12 3P
00 Q
11 P
01 P
2
11 Q
04 +
3Q
11 P
01 P
2
00 Q
04 P
22 +3Q
11 P
01 P
2
00 Q
24 P
02 P
00 Q
31 Q
04 P
3
01 +
2Q
20 P
2
01 Q
02 P
02 P
12 3P
00 Q
11 P
2
01 Q
04 P
21 +3Q
31 P
01 P
2
00 Q
04 P
02 +
3Q
11 P
21 P
2
00 Q
04 P
02 +2Q
00 P
01 P
11 Q
02 P
2
12 8Q
20 P
01 P
11 P
00 Q
04 P
02 +
4Q
20 P
10 P
2
02 P
00 Q
04 P
00 Q
11 P
21 Q
02 P
2
02 +6P
00 Q
11 P
11 P
10 Q
04 P
02
2P
00 Q
11 P
11 Q
02 P
02 P
12 4Q
20 P
2
01 P
00 Q
04 P
12 Q
20 P
3
01 Q
13 P
02 +
2P
00 Q
11 P
01 P
11 Q
13 P
02 P
00 Q
11 Q
24 P
3
01 Q
20 P
01 Q
11 P
3
02 +
3P
00 Q
2
11 P
2
02 P
4
R
4
= Q
20 Q
24 P
4
01 +P
2
20 Q
2
02 P
2
02 +Q
2
00 P
4
12 +
2Q
00 Q
40 P
4
02 +Q
2
20 P
4
02 +2P
4
00 Q
04 Q
44 +
P 4
00 Q
2
24 4Q
20 P
2
01 P
00 Q
24 P
02 +8P
3
00 P
20 Q
04 Q
24
6P 2
00 Q
02 P
20 Q
24 P
02 2P
00 P
10 Q
02 P
21 Q
13 P
02 2P
00 P
10 Q
02 P
01 Q
13 P
22
Q
00 P
01 Q
11 P
3
12 6Q
00 P
01 Q
11 P
02 P
12 P
22 3Q
00 P
01 Q
31 P
2
02 P
12 +
Q
00 P
2
01 Q
22 P
2
12 +6P
00 Q
11 P
21 P
10 Q
04 P
02 +6P
00 Q
11 P
11 P
20 Q
04 P
02
3P
00 Q
11 P
2
01 Q
24 P
11 P
00 Q
11 P
11 Q
22 P
2
02 P
00 Q
11 P
11 Q
02 P
2
12 +
P
00 Q
11 P
2
01 Q
33 P
02 3Q
20 P
01 Q
11 P
2
02 P
12 +6P
00 Q
11 P
01 P
10 Q
24 P
02
6P
00 Q
11 P
01 P
11 Q
04 P
21 +2P
00 Q
11 P
01 P
21 Q
13 P
02 4Q
20 P
2
01 P
00 Q
04 P
22
2P
00 Q
11 P
01 Q
22 P
02 P
12 8Q
20 P
01 P
21 P
00 Q
04 P
02 8Q
20 P
01 P
11 P
10 Q
04 P
02 +
P 4
10 Q
2
04 2P
00 Q
11 P
21 Q
02 P
02 P
12 +2Q
20 P
2
01 Q
02 P
02 P
22 +
P
00 Q
11 P
2
11 Q
13 P
02 3Q
20 P
2
01 P
11 Q
13 P
02 +6P
00 Q
11 P
01 P
10 Q
04 P
22 +
6P
00 Q
11 P
01 P
20 Q
04 P
12 +2P
00 Q
11 Q
31 P
3
02 4Q
40 P
2
01 P
00 Q
04 P
02
8Q
20 P
01 P
11 P
00 Q
04 P
12 +2Q
20 P
01 P
21 Q
02 P
2
02 +6Q
2
00 P
2
02 P
2
22 +
4Q
20 P
01 P
11 Q
02 P
02 P
12 +Q
40 P
2
01 Q
02 P
2
02 4Q
20 P
2
11 P
00 Q
04 P
02 +
3Q
20 P
11 P
00 P
2
02 Q
13 2Q
20 P
20 P
3
02 Q
02 +4Q
20 P
2
00 P
02 Q
04 P
22
6Q
20 P
10 P
2
02 Q
02 P
12 +8Q
20 P
10 P
02 P
00 Q
04 P
12 +6Q
20 P
01 P
00 P
02 Q
13 P
12
6Q
20 P
00 P
02 Q
02 P
2
12 +3Q
20 P
01 P
10 P
2
02 Q
13 6Q
20 P
00 P
2
02 Q
02 P
22 +
2Q
20 P
2
00 P
2
02 Q
24 +P
2
00 Q
2
02 P
2
22 +Q
00 P
2
01 Q
02 P
2
22 +
6Q
00 P
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