Here we list the multi-variable formulas for the 21 examples in the article. 

QPochhammer[a, q, n] means $(a,q)_n$; QPochhammer[a, q] means $(a,q)_\infty$; 

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Example I:

Sum[((-1)^n q^(b n + c n + 7 n^2) (1 - q^(4 + 2 b + 8 n) - q^(4 + 2 c + 8 n) - q^(1 + b + c + 10 n) - q^(3 + b + c + 10 n) + q^(4 + 2 b + 12 n) + q^(4 + 2 c + 12 n) - q^(5 + b + c + 14 n) + q^(4 + 2 b + 2 c + 16 n) + q^(8 + 2 b + 2 c + 16 n) + q^(6 + 2 b + 2 c + 20 n) - q^(9 + 3 b + 3 c + 26 n)) QPochhammer[q^(1 + b - c), q^2, n] QPochhammer[q^(1 - b + c), q^2, n] QPochhammer[q^(1 + b + c), q^2, 3 n])/((1 - q^(4 + 2 b + 8 n)) (1 - q^(4 + 2 c + 8 n)) QPochhammer[q^4, q^4, n] QPochhammer[q^(4 + 2 b), q^4, 2 n] QPochhammer[q^(4 + 2 c), q^4, 2 n]), {n, 0, Infinity}]

= QPochhammer[q^(1 + b + c), q^2]/(QPochhammer[q^(4 + 2 b), q^4]QPochhammer[q^(4 + 2 c), q^4])
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Example II:

Sum[(q^(b n + d n + 2 n^2) (q^(2 c) - q^(2 + 2 b + d + 4 n) - q^(1 + b + c + d + 4 n) - q^(1 + 2 b + c + d + 4 n) + q^(2 + 2 b + d + 6 n) + q^(2 + 3 b + 2 d + 6 n)) QPochhammer[q^(1 - c), q^2, n] QPochhammer[q^(1 + b - c), q^2, n] QPochhammer[q^(1 + b - c + d), q^2, n] QPochhammer[q^(1 + 2 b - c + d), q^2, n])/((q^(2 c) - q^(2 + 2 b + d + 4 n)) QPochhammer[q^2, q^2, n] QPochhammer[q^(2 + b + d), q^2, n] QPochhammer[q^(2 + 2 b - 2 c + d), q^2, 2 n]), {n, 0, Infinity}]

= (QPochhammer[q^(1 + b - c + d), q^2] QPochhammer[q^(1 + 2 b - c + d), q^2])/(QPochhammer[q^(2 + b + d), q^2] QPochhammer[q^(2 + 2 b - 2 c + d), q^2])
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Example III:

Sum[(q^(c n + 2 d n + 6 n^2) (1 - q^(2 + 4 n) - q^(2 + d + 4 n) - q^(2 + c + d + 4 n) - q^(1 + b + d + 6 n) - q^(1 - b + c + d + 6 n) + q^(2 + d + 8 n) + q^(4 + d + 8 n) + q^(2 + c + d + 8 n) + q^(4 + c + d + 8 n) + q^(2 + c + 2 d + 8 n) + q^(4 + c + 2 d + 8 n) + q^(4 + 2 b + 2 d + 12 n) - q^(6 + c + 2 d + 12 n) + q^(4 - 2 b + 2 c + 2 d + 12 n) - q^(5 + b + 2 d + 14 n) - q^(5 - b + c + 2 d + 14 n) - q^(5 + b + c + 2 d + 14 n) - q^(5 - b + 2 c + 2 d + 14 n) - q^(5 + b + c + 3 d + 14 n) - q^(5 - b + 2 c + 3 d + 14 n) + q^(6 + c + 2 d + 16 n) + q^(6 + c + 3 d + 16 n) + q^(6 + 2 c + 3 d + 16 n)) QPochhammer[q^(1 - b), q^2, n] QPochhammer[q^(1 + b), q^2, n] QPochhammer[q^(1 + b - c), q^2, n] QPochhammer[q^(1 - b + c), q^2, n] QPochhammer[q^(1 + b + d), q^2, n] QPochhammer[q^(1 - b + c + d), q^2, n])/((1 - q^(2 + 4 n)) (1 - q^(2 + d + 4 n)) (1 - q^(2 + c + d + 4 n)) QPochhammer[q^2, q^2, 2 n] QPochhammer[q^(2 + d), q^2, 2 n] QPochhammer[q^(2 + c + d), q^2, 2 n]), {n, 0, Infinity}]

= (QPochhammer[q^(1 + b + d), q^2] QPochhammer[q^(1 - b + c + d), q^2])/(QPochhammer[q^(2 + d), q^2] QPochhammer[q^(2 + c + d), q^2])
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Example IV:

Sum[((-1)^n q^(b n + 2 c n + 3 n^2) (1 - q^(1 + b + 2 c + 6 n)) QPochhammer[q^(1 - b), q^2, n] QPochhammer[q^(1 + b), q^2, n] QPochhammer[q^(1 + b + 2 c), q^2, n])/(QPochhammer[q^4, q^4, n] QPochhammer[q^(4 + 2 c), q^4, n] QPochhammer[q^(4 + 2 b + 2 c), q^4, n]), {n, 0, Infinity}]

= QPochhammer[q^(1 + b + 2 c), q^2]/(QPochhammer[q^(4 + 2 c), q^4] QPochhammer[q^(4 + 2 b + 2 c), q^4])
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Example V:

Sum[((-1)^n q^(b n + 2 d n + 3 n^2) (1 + q^(4 - 2 b + 2 c + 2 d + 12 n) - q^(5 - b + 2 c + 2 d + 14 n) + q^(2 + c + d + 8 n) (1 + q^2 - q^(1 - b + d)) + q^(1 + d + 6 n) (q - q^(-b + c) + q^(1 + c + d)) - q^(1 + 4 n) (q + q^(b + d) + q^(1 + c + d))) QPochhammer[q^(1 - b), q^2, n] QPochhammer[q^(1 + b), q^2, n] QPochhammer[q^(1 - b + c), q^2, 2 n] QPochhammer[q^(1 - b + c + d), q^2, n])/((1 - q^(2 + 4 n)) (1 - q^(2 + c + d + 4 n)) QPochhammer[q^2, q^2, 2 n] QPochhammer[q^(2 + d), q^2, n] QPochhammer[q^(2 + c + d), q^2, 2 n]), {n, 0, Infinity}]

= (QPochhammer[q^(1 + b + d), q^2] QPochhammer[q^(1 - b + c + d), q^2])/(QPochhammer[q^(2 + d), q^2] QPochhammer[q^(2 + c + d), q^2])
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Example VI:

Sum[((-1)^n q^(2 d n + 4 e n + 2 n^2) (1 - q^(3 + 2 b + 4 e + 8 n) - q^(2 + 2 c + 4 e + 8 n) - q^(4 + 2 b + 2 c + 4 e + 8 n) - q^(1 + 2 b + 2 c + 2 d + 4 e + 8 n) - q^(2 + 2 b + 2 c + 2 d + 8 e + 8 n) + q^(4 + 2 b + 2 c + 4 e + 12 n) + q^(5 + 2 b + 2 c + 8 e + 12 n) + q^(2 + 2 b + 2 c + 2 d + 8 e + 12 n) + q^(4 + 4 b + 2 c + 2 d + 8 e + 12 n) + q^(3 + 2 b + 4 c + 2 d + 8 e + 12 n) - q^(6 + 4 b + 4 c + 2 d + 12 e + 20 n)) QPochhammer[q^(2 + 2 b), q^4, n] QPochhammer[q^(1 + 2 c), q^4, n] QPochhammer[q^(3 - 2 d), q^4, n] QPochhammer[q^(3 + 2 b + 4 e), q^4, n] QPochhammer[q^(2 + 2 c + 4 e), q^4, n] QPochhammer[q^(1 + 2 b + 2 c + 2 d + 4 e), q^4, n])/((1 - q^(1 + 4 e + 4 n)) (1 - q^(4 + 2 b + 2 c + 4 e + 8 n)) QPochhammer[q^4, q^4, n] QPochhammer[q^(1 + 4 e), q^4, n] QPochhammer[q^(4 + 2 b + 2 c + 4 e), q^4, 2 n] QPochhammer[q^(4 + 2 b + 2 d + 4 e), q^4, n] QPochhammer[q^(3 + 2 c + 2 d + 4 e), q^4, n]), {n, 0, Infinity}]

= (QPochhammer[q^(3 + 2 b + 4 e), q^4] QPochhammer[q^(2 + 2 c + 4 e), q^4] QPochhammer[q^(2 + 2 d + 4 e), q^4] QPochhammer[q^(1 + 2 b + 2 c + 2 d + 4 e), q^4])/(QPochhammer[q^(1 + 4 e), q^4] QPochhammer[q^(4 + 2 b + 2 c + 4 e), q^4] QPochhammer[q^(4 + 2 b + 2 d + 4 e), q^4] QPochhammer[q^(3 + 2 c + 2 d + 4 e), q^4])
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Example VII:

Sum[((-1)^n q^(b n + c n + d n + 2 e n + n^2) (1 - q^(2 + 4 n) - q^(1 + b + e + 4 n) - q^(1 + c + e + 4 n) - q^(1 + d + e + 4 n) - q^(1 + b + c + d + e + 4 n) + q^(2 + e + 6 n) + q^(2 + b + c + e + 6 n) + q^(2 + b + d + e + 6 n) + q^(2 + c + d + e + 6 n) + q^(1 + b + c + d + 2 e + 6 n) - q^(3 + b + c + d + 2 e + 10 n)) QPochhammer[q^(1 - b), q^2, n] QPochhammer[q^(1 + b), q^2, n] QPochhammer[q^(1 - c), q^2, n] QPochhammer[q^(1 + c), q^2, n] QPochhammer[q^(1 - d), q^2, n] QPochhammer[q^(1 + d), q^2, n])/((1 - q^(2 + 4 n)) QPochhammer[q^2, q^2, 2 n] QPochhammer[q^(2 + e), q^2, n] QPochhammer[q^(2 + b + c + e), q^2, n] QPochhammer[q^(2 + b + d + e), q^2, n] QPochhammer[q^(2 + c + d + e), q^2, n]), {n, 0, Infinity}]

= (QPochhammer[q^(1 + b + e), q^2] QPochhammer[q^(1 + c + e), q^2] QPochhammer[q^(1 + d + e), q^2] QPochhammer[q^(1 + b + c + d + e), q^2])/(QPochhammer[q^(2 + e), q^2] QPochhammer[q^(2 + b + c + e), q^2] QPochhammer[q^(2 + b + d + e), q^2] QPochhammer[q^(2 + c + d + e), q^2])
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Example VIII:

Sum[((-1)^n q^(b n + c n + d n + 2 e n + 5 n^2) (1 - q^(2 + 4 n) - q^(2 + e + 4 n) - q^(2 + b + c + e + 4 n) - q^(2 + b + d + e + 4 n) - q^(2 + c + d + e + 4 n) - q^(1 + b + e + 6 n) - q^(1 + c + e + 6 n) - q^(1 + d + e + 6 n) - q^(1 + b + c + d + e + 6 n) - q^(1 + b + c + d + 2 e + 6 n) + q^(2 + e + 8 n) + q^(4 + e + 8 n) + q^(2 + b + c + e + 8 n) + q^(4 + b + c + e + 8 n) + q^(2 + b + d + e + 8 n) + q^(4 + b + d + e + 8 n) + q^(2 + c + d + e + 8 n) + q^(4 + c + d + e + 8 n) + q^(2 + b + c + 2 e + 8 n) + q^(4 + b + c + 2 e + 8 n) + q^(2 + b + d + 2 e + 8 n) + q^(4 + b + d + 2 e + 8 n) + q^(2 + c + d + 2 e + 8 n) + q^(4 + c + d + 2 e + 8 n) + q^(2 + 2 b + c + d + 2 e + 8 n) + q^(4 + 2 b + c + d + 2 e + 8 n) + q^(2 + b + 2 c + d + 2 e + 8 n) + q^(4 + b + 2 c + d + 2 e + 8 n) + q^(2 + b + c + 2 d + 2 e + 8 n) + q^(4 + b + c + 2 d + 2 e + 8 n) + q^(1 + b + c + d + 2 e + 10 n) + q^(3 + b + c + d + 2 e + 10 n) + q^(4 + 2 b + 2 e + 12 n) - q^(6 + b + c + 2 e + 12 n) + q^(4 + 2 c + 2 e + 12 n) - q^(6 + b + d + 2 e + 12 n) - q^(6 + c + d + 2 e + 12 n) - q^(6 + 2 b + c + d + 2 e + 12 n) - q^(6 + b + 2 c + d + 2 e + 12 n) + q^(4 + 2 d + 2 e + 12 n) - q^(6 + b + c + 2 d + 2 e + 12 n) + q^(4 + 2 b + 2 c + 2 d + 2 e + 12 n) - q^(6 + 2 b + c + d + 3 e + 12 n) - q^(6 + b + 2 c + d + 3 e + 12 n) - q^(6 + b + c + 2 d + 3 e + 12 n) - q^(6 + 2 b + 2 c + 2 d + 3 e + 12 n) + q^(4 + 2 b + 2 c + 2 d + 4 e + 12 n) - q^(5 + b + 2 e + 14 n) - q^(5 + c + 2 e + 14 n) - q^(5 + 2 b + c + 2 e + 14 n) - q^(5 + b + 2 c + 2 e + 14 n) - q^(5 + d + 2 e + 14 n) - q^(5 + 2 b + d + 2 e + 14 n) - q^(3 + b + c + d + 2 e + 14 n) - 3 q^(5 + b + c + d + 2 e + 14 n) - q^(5 + 2 c + d + 2 e + 14 n) - q^(5 + 2 b + 2 c + d + 2 e + 14 n) - q^(5 + b + 2 d + 2 e + 14 n) - q^(5 + c + 2 d + 2 e + 14 n) - q^(5 + 2 b + c + 2 d + 2 e + 14 n) - q^(5 + b + 2 c + 2 d + 2 e + 14 n) - q^(5 + 2 b + c + 3 e + 14 n) - q^(5 + b + 2 c + 3 e + 14 n) - q^(5 + 2 b + d + 3 e + 14 n) - q^(3 + b + c + d + 3 e + 14 n) - 3 q^(5 + b + c + d + 3 e + 14 n) - q^(5 + 3 b + c + d + 3 e + 14 n) - q^(5 + 2 c + d + 3 e + 14 n) - q^(3 + 2 b + 2 c + d + 3 e + 14 n) - 3 q^(5 + 2 b + 2 c + d + 3 e + 14 n) - q^(5 + b + 3 c + d + 3 e + 14 n) - q^(5 + b + 2 d + 3 e + 14 n) - q^(5 + c + 2 d + 3 e + 14 n) - q^(3 + 2 b + c + 2 d + 3 e + 14 n) - 3 q^(5 + 2 b + c + 2 d + 3 e + 14 n) - q^(3 + b + 2 c + 2 d + 3 e + 14 n) - 3 q^(5 + b + 2 c + 2 d + 3 e + 14 n) - q^(5 + 3 b + 2 c + 2 d + 3 e + 14 n) - q^(5 + 2 b + 3 c + 2 d + 3 e + 14 n) - q^(5 + b + c + 3 d + 3 e + 14 n) - q^(5 + 2 b + 2 c + 3 d + 3 e + 14 n) - q^(5 + 2 b + 2 c + d + 4 e + 14 n) - q^(5 + 2 b + c + 2 d + 4 e + 14 n) - q^(5 + b + 2 c + 2 d + 4 e + 14 n) - q^(5 + 3 b + 2 c + 2 d + 4 e + 14 n) - q^(5 + 2 b + 3 c + 2 d + 4 e + 14 n) - q^(5 + 2 b + 2 c + 3 d + 4 e + 14 n) + q^(6 + b + c + 2 e + 16 n) + q^(6 + b + d + 2 e + 16 n) + q^(6 + c + d + 2 e + 16 n) + q^(6 + 2 b + c + d + 2 e + 16 n) + q^(6 + b + 2 c + d + 2 e + 16 n) + q^(6 + b + c + 2 d + 2 e + 16 n) + q^(6 + b + c + 3 e + 16 n) + q^(6 + 2 b + 2 c + 3 e + 16 n) + q^(6 + b + d + 3 e + 16 n) + q^(6 + c + d + 3 e + 16 n) + 3 q^(6 + 2 b + c + d + 3 e + 16 n) + q^(8 + 2 b + c + d + 3 e + 16 n) + 3 q^(6 + b + 2 c + d + 3 e + 16 n) + q^(8 + b + 2 c + d + 3 e + 16 n) + q^(6 + 3 b + 2 c + d + 3 e + 16 n) + q^(6 + 2 b + 3 c + d + 3 e + 16 n) + q^(6 + 2 b + 2 d + 3 e + 16 n) + 3 q^(6 + b + c + 2 d + 3 e + 16 n) + q^(8 + b + c + 2 d + 3 e + 16 n) + q^(6 + 3 b + c + 2 d + 3 e + 16 n) + q^(6 + 2 c + 2 d + 3 e + 16 n) + 3 q^(6 + 2 b + 2 c + 2 d + 3 e + 16 n) + q^(8 + 2 b + 2 c + 2 d + 3 e + 16 n) + q^(6 + b + 3 c + 2 d + 3 e + 16 n) + q^(6 + 2 b + c + 3 d + 3 e + 16 n) + q^(6 + b + 2 c + 3 d + 3 e + 16 n) + q^(6 + 2 b + c + d + 4 e + 16 n) + q^(6 + b + 2 c + d + 4 e + 16 n) + q^(6 + 3 b + 2 c + d + 4 e + 16 n) + q^(6 + 2 b + 3 c + d + 4 e + 16 n) + q^(6 + b + c + 2 d + 4 e + 16 n) + q^(6 + 3 b + c + 2 d + 4 e + 16 n) + 3 q^(6 + 2 b + 2 c + 2 d + 4 e + 16 n) + q^(8 + 2 b + 2 c + 2 d + 4 e + 16 n) + q^(6 + b + 3 c + 2 d + 4 e + 16 n) + q^(6 + 3 b + 3 c + 2 d + 4 e + 16 n) + q^(6 + 2 b + c + 3 d + 4 e + 16 n) + q^(6 + b + 2 c + 3 d + 4 e + 16 n) + q^(6 + 3 b + 2 c + 3 d + 4 e + 16 n) + q^(6 + 2 b + 3 c + 3 d + 4 e + 16 n) - q^(7 + b + c + d + 2 e + 18 n) + q^(5 + b + c + d + 3 e + 18 n) + q^(5 + 2 b + 2 c + d + 3 e + 18 n) + q^(5 + 2 b + c + 2 d + 3 e + 18 n) + q^(5 + b + 2 c + 2 d + 3 e + 18 n) - q^(7 + b + c + d + 4 e + 18 n) + q^(5 + 2 b + 2 c + d + 4 e + 18 n) - q^(7 + 3 b + 3 c + d + 4 e + 18 n) + q^(5 + 2 b + c + 2 d + 4 e + 18 n) + q^(5 + b + 2 c + 2 d + 4 e + 18 n) + q^(5 + 3 b + 2 c + 2 d + 4 e + 18 n) + q^(5 + 2 b + 3 c + 2 d + 4 e + 18 n) - q^(7 + 3 b + c + 3 d + 4 e + 18 n) + q^(5 + 2 b + 2 c + 3 d + 4 e + 18 n) - q^(7 + b + 3 c + 3 d + 4 e + 18 n) - q^(8 + 2 b + 2 c + 2 d + 4 e + 20 n) - q^(10 + 2 b + 2 c + 2 d + 4 e + 20 n) - q^(7 + 2 b + 2 c + d + 4 e + 22 n) - q^(9 + 2 b + 2 c + d + 4 e + 22 n) - q^(7 + 2 b + c + 2 d + 4 e + 22 n) - q^(9 + 2 b + c + 2 d + 4 e + 22 n) - q^(7 + b + 2 c + 2 d + 4 e + 22 n) - q^(9 + b + 2 c + 2 d + 4 e + 22 n) - q^(7 + 3 b + 2 c + 2 d + 4 e + 22 n) - q^(9 + 3 b + 2 c + 2 d + 4 e + 22 n) - q^(7 + 2 b + 3 c + 2 d + 4 e + 22 n) - q^(9 + 2 b + 3 c + 2 d + 4 e + 22 n) - q^(7 + 2 b + 2 c + 3 d + 4 e + 22 n) - q^(9 + 2 b + 2 c + 3 d + 4 e + 22 n) - q^(7 + 3 b + 2 c + 2 d + 5 e + 22 n) - q^(9 + 3 b + 2 c + 2 d + 5 e + 22 n) - q^(7 + 2 b + 3 c + 2 d + 5 e + 22 n) - q^(9 + 2 b + 3 c + 2 d + 5 e + 22 n) - q^(7 + 2 b + 2 c + 3 d + 5 e + 22 n) - q^(9 + 2 b + 2 c + 3 d + 5 e + 22 n) - q^(7 + 3 b + 3 c + 3 d + 5 e + 22 n) - q^(9 + 3 b + 3 c + 3 d + 5 e + 22 n) + q^(10 + 2 b + 2 c + 2 d + 4 e + 24 n) + q^(10 + 2 b + 2 c + 2 d + 5 e + 24 n) + q^(10 + 3 b + 3 c + 2 d + 5 e + 24 n) + q^(10 + 3 b + 2 c + 3 d + 5 e + 24 n) + q^(10 + 2 b + 3 c + 3 d + 5 e + 24 n) + q^(9 + 3 b + 2 c + 2 d + 5 e + 26 n) + q^(9 + 2 b + 3 c + 2 d + 5 e + 26 n) + q^(9 + 2 b + 2 c + 3 d + 5 e + 26 n) + q^(9 + 3 b + 3 c + 3 d + 5 e + 26 n) + q^(9 + 3 b + 3 c + 3 d + 6 e + 26 n) - q^(11 + 3 b + 3 c + 3 d + 6 e + 30 n)) QPochhammer[q^(1 - b), q^2, n] QPochhammer[q^(1 + b), q^2, n] QPochhammer[q^(1 - c), q^2, n] QPochhammer[q^(1 + c), q^2, n] QPochhammer[q^(1 - d), q^2, n] QPochhammer[q^(1 + d), q^2, n] QPochhammer[q^(1 + b + e), q^2, n] QPochhammer[q^(1 + c + e), q^2, n] QPochhammer[q^(1 + d + e), q^2, n] QPochhammer[q^(1 + b + c + d + e), q^2, n])/((1 - q^(2 + 4 n)) (1 - q^(2 + e + 4 n)) (1 - q^(2 + b + c + e + 4 n)) (1 - q^(2 + b + d + e + 4 n)) (1 - q^(2 + c + d + e + 4 n)) QPochhammer[q^2, q^2, 2 n] QPochhammer[q^(2 + e), q^2, 2 n] QPochhammer[q^(2 + b + c + e), q^2, 2 n] QPochhammer[q^(2 + b + d + e), q^2, 2 n] QPochhammer[q^(2 + c + d + e), q^2, 2 n]), {n, 0, Infinity}]

= (QPochhammer[q^(1 + b + e), q^2] QPochhammer[q^(1 + c + e), q^2] QPochhammer[q^(1 + d + e), q^2] QPochhammer[q^(1 + b + c + d + e), q^2])/(QPochhammer[q^(2 + e), q^2] QPochhammer[q^(2 + b + c + e), q^2] QPochhammer[q^(2 + b + d + e), q^2] QPochhammer[q^(2 + c + d + e), q^2])
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Example IX:

Sum[(q^(b n + c n + e n + 2 n^2) (1 - q^(2 + e + 4 n) - q^(1 + b + e + 4 n) - q^(1 + c + e + 4 n) - q^(2 + b + d + e + 4 n) - q^(2 + c + d + e + 4 n) - q^(1 + b + c + d + e + 4 n) + q^(2 + e + 6 n) - q^(1 + d + e + 6 n) + q^(2 + b + d + e + 6 n) + q^(2 + c + d + e + 6 n) + q^(2 + b + c + 2 e + 6 n) - q^(1 + b + c + d + 2 e + 6 n) + q^(2 + 2 b + c + d + 2 e + 6 n) + q^(2 + b + 2 c + d + 2 e + 6 n) - q^(3 + d + e + 8 n) + q^(2 + b + d + 2 e + 8 n) + q^(4 + b + d + 2 e + 8 n) + q^(2 + c + d + 2 e + 8 n) + q^(4 + c + d + 2 e + 8 n) + q^(1 + b + c + d + 2 e + 8 n) + q^(2 + b + c + 2 d + 2 e + 8 n) + q^(4 + b + c + 2 d + 2 e + 8 n) - q^(3 + 2 b + 2 c + d + 3 e + 8 n) + q^(4 + 2 d + 2 e + 12 n) - q^(3 + b + c + d + 3 e + 12 n) - q^(5 + b + c + d + 3 e + 12 n) - q^(6 + b + c + 2 d + 3 e + 12 n) - q^(3 + 2 b + c + 2 d + 3 e + 12 n) - q^(5 + 2 b + c + 2 d + 3 e + 12 n) - q^(3 + b + 2 c + 2 d + 3 e + 12 n) - q^(5 + b + 2 c + 2 d + 3 e + 12 n) + q^(4 + 2 b + 2 c + 2 d + 4 e + 12 n) - q^(5 + b + 2 d + 3 e + 14 n) - q^(5 + c + 2 d + 3 e + 14 n) + q^(6 + b + c + 2 d + 3 e + 14 n) - q^(5 + b + c + 3 d + 3 e + 14 n) - q^(5 + 2 b + c + 2 d + 4 e + 14 n) - q^(5 + b + 2 c + 2 d + 4 e + 14 n) + q^(6 + 2 b + 2 c + 2 d + 4 e + 14 n) - q^(5 + 2 b + 2 c + 3 d + 4 e + 14 n) + q^(6 + b + c + 2 d + 4 e + 16 n) + q^(5 + 2 b + c + 2 d + 4 e + 16 n) + q^(5 + b + 2 c + 2 d + 4 e + 16 n) + q^(6 + 2 b + c + 3 d + 4 e + 16 n) + q^(6 + b + 2 c + 3 d + 4 e + 16 n) + q^(5 + 2 b + 2 c + 3 d + 4 e + 16 n) - q^(7 + 2 b + 2 c + 3 d + 5 e + 20 n)) QPochhammer[q^(1 - b), q^2, n] QPochhammer[q^(1 - c), q^2, n] QPochhammer[q^(1 + d), q^2, n] QPochhammer[q^(1 + b + e), q^2, n] QPochhammer[q^(1 + c + e), q^2, n] QPochhammer[q^(1 + d + e), q^2, 2 n] QPochhammer[q^(1 + b + c + d + e), q^2, n])/((1 - q^(2 + e + 4 n)) (1 - q^(2 + b + d + e + 4 n)) (1 - q^(2 + c + d + e + 4 n)) QPochhammer[q^2, q^2, n] QPochhammer[q^(2 + e), q^2, 2 n] QPochhammer[q^(2 + b + c + e), q^2, n] QPochhammer[q^(2 + b + d + e), q^2, 2 n] QPochhammer[q^(2 + c + d + e), q^2, 2 n]), {n, 0, Infinity}]

= (QPochhammer[q^(1 + b + e), q^2] QPochhammer[q^(1 + c + e), q^2] QPochhammer[q^(1 + d + e), q^2] QPochhammer[q^(1 + b + c + d + e), q^2])/(QPochhammer[q^(2 + e), q^2] QPochhammer[q^(2 + b + c + e), q^2] QPochhammer[q^(2 + b + d + e), q^2] QPochhammer[q^(2 + c + d + e), q^2])
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Example X:

Sum[((-1)^n q^(b n + c n + 5 n^2) (1 - q^(4 + 8 n) - q^(1 + b + c + 10 n) - q^(3 + b + c + 10 n) - q^(3 - b + 12 n) - q^(3 + b + 12 n) + q^(4 + c + 14 n) + q^(4 + 2 b + c + 14 n) + q^(4 + 16 n) + q^(6 + 16 n) + q^(3 + b + c + 18 n) - q^(7 + b + c + 26 n)) QPochhammer[q^(1 - b), q^2, 2 n] QPochhammer[q^(1 + b), q^2, 2 n] QPochhammer[q^(3 - b - c), q^6, n] QPochhammer[q^(1 + b + c), q^2, n])/((1 - q^(2 + 8 n) - q^(4 + 8 n) + q^(6 + 16 n)) QPochhammer[q^2, q^2, 4 n] QPochhammer[q^(6 + c), q^6, n] QPochhammer[q^(6 + 2 b + c), q^6, n]), {n, 0, Infinity}]

= QPochhammer[q^6, q^6] QPochhammer[q^(3 - b), q^6] QPochhammer[q^(3 + b), q^6] QPochhammer[q^(1 + b + c), q^2])/(QPochhammer[q^2, q^2] QPochhammer[q^(6 + c), q^6] QPochhammer[q^(3 + b + c), q^6] QPochhammer[q^(6 + 2 b + c), q^6])
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Example XI:

Sum[(q^(-2 b n + c n + n^2) (1 - q^(1 + c + 2 d + 6 n)) QPochhammer[q^(1 - c), q^2, n] QPochhammer[q^(1 + c), q^2, n] QPochhammer[q^(2 + 2 b + 2 d), q^4, n] QPochhammer[q^(1 + c + 2 d), q^2, n])/(QPochhammer[q^4, q^4, n] QPochhammer[q^(1 - 2 b + c), q^2, n] QPochhammer[q^(4 + 2 d), q^4, n] QPochhammer[q^(4 + 2 c + 2 d), q^4, n]), {n, 0, Infinity}]

= (QPochhammer[q^(2 - 2 b), q^4] QPochhammer[q^(2 - 2 b + 2 c), q^4] QPochhammer[q^(1 + c + 2 d), q^2])/(QPochhammer[q^(1 - 2 b + c), q^2] QPochhammer[q^(4 + 2 d), q^4] QPochhammer[q^(4 + 2 c + 2 d), q^4])
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Example XII:

Sum[(q^(-2 b n + c n + 5 n^2) (1 - q^(3 - 2 b + c + 6 n) - q^(2 - 2 b + 8 n) - q^(2 - 2 b + 2 c + 8 n) - q^(4 + 2 d + 8 n) - q^(4 + 2 c + 2 d + 8 n) + q^(3 - 2 b + c + 10 n) - q^(1 + c + 2 d + 10 n) - q^(3 + c + 2 d + 10 n) + q^(4 - 4 b + 2 c + 12 n) + q^(4 + 2 d + 12 n) + q^(4 + 2 c + 2 d + 12 n) - q^(2 - 2 b + 2 c + 2 d + 12 n) - q^(5 + c + 2 d + 14 n) + q^(3 - 2 b + c + 2 d + 14 n) + q^(5 - 2 b + c + 2 d + 14 n) + q^(7 - 2 b + c + 2 d + 14 n) + q^(3 - 2 b + 3 c + 2 d + 14 n) + q^(5 - 2 b + 3 c + 2 d + 14 n) + q^(7 - 2 b + 3 c + 2 d + 14 n) + q^(2 - 2 b + 2 c + 2 d + 16 n) + q^(4 + 2 c + 4 d + 16 n) + q^(8 + 2 c + 4 d + 16 n) - q^(7 - 4 b + 3 c + 2 d + 18 n) + q^(6 + 2 c + 4 d + 20 n) - q^(5 - 4 b + 3 c + 2 d + 22 n) - q^(9 - 4 b + 3 c + 2 d + 22 n) - q^(11 - 2 b + 3 c + 4 d + 22 n) - q^(6 - 2 b + 2 c + 4 d + 24 n) - q^(8 - 2 b + 2 c + 4 d + 24 n) - q^(10 - 2 b + 2 c + 4 d + 24 n) + q^(8 - 4 b + 4 c + 4 d + 24 n) - q^(6 - 2 b + 4 c + 4 d + 24 n) - q^(8 - 2 b + 4 c + 4 d + 24 n) - q^(10 - 2 b + 4 c + 4 d + 24 n) - q^(9 - 4 b + 3 c + 4 d + 26 n) + q^(11 - 2 b + 3 c + 4 d + 26 n) - q^(9 - 4 b + 5 c + 4 d + 26 n) - q^(9 + 3 c + 6 d + 26 n) + q^(10 - 4 b + 4 c + 4 d + 28 n) + q^(12 - 4 b + 4 c + 4 d + 28 n) - q^(10 - 2 b + 4 c + 6 d + 28 n) + q^(9 - 4 b + 3 c + 4 d + 30 n) + q^(9 - 4 b + 5 c + 4 d + 30 n) + q^(11 - 2 b + 3 c + 6 d + 30 n) + q^(11 - 2 b + 5 c + 6 d + 30 n) + q^(10 - 2 b + 4 c + 6 d + 32 n) - q^(13 - 4 b + 5 c + 6 d + 38 n)) QPochhammer[q^(2 - 2 b), q^4, n] QPochhammer[q^(1 - c), q^2, n] QPochhammer[q^(1 + c), q^2, n] QPochhammer[q^(2 - 2 b + 2 c), q^4, n] QPochhammer[q^(2 + 2 b + 2 d), q^4, n] QPochhammer[q^(1 + c + 2 d), q^2, 3 n])/((1 - q^(1 - 2 b + c + 6 n)) (1 - q^(3 - 2 b + c + 6 n)) (1 - q^(4 + 2 d + 8 n)) (1 - q^(4 + 2 c + 2 d + 8 n)) QPochhammer[q^4, q^4, n] QPochhammer[q^(1 - 2 b + c), q^2, 3 n] QPochhammer[q^(4 + 2 d), q^4, 2 n] QPochhammer[q^(4 + 2 c + 2 d), q^4, 2 n]), {n, 0, Infinity}]

= (QPochhammer[q^(2 - 2 b), q^4] QPochhammer[q^(2 - 2 b + 2 c), q^4] QPochhammer[q^(1 + c + 2 d), q^2])/(QPochhammer[q^(1 - 2 b + c), q^2] QPochhammer[q^(4 + 2 d), q^4] QPochhammer[q^(4 + 2 c + 2 d), q^4])
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Example XIII:

Sum[(q^(4 n^2) (1 + q^(4 n) + 9 q^(1 + 2 n) + 36 q^(2 + 4 n) - 7 q^(1 + 6 n) + 84 q^(3 + 6 n) - 63 q^(2 + 8 n) + 126 q^(4 + 8 n) - 231 q^(3 + 10 n) + 126 q^(5 + 10 n) + 35 q^(2 + 12 n) - 448 q^(4 + 12 n) + 84 q^(6 + 12 n) + 187 q^(3 + 14 n) - 560 q^(5 + 14 n) + 36 q^(7 + 14 n) - 9 q^(2 + 16 n) + 509 q^(4 + 16 n) - 441 q^(6 + 16 n) + 9 q^(8 + 16 n) - 65 q^(3 + 18 n) + 805 q^(5 + 18 n) - 225 q^(7 + 18 n) + q^(9 + 18 n) + q^(2 + 20 n) - 225 q^(4 + 20 n) + 805 q^(6 + 20 n) - 65 q^(8 + 20 n) + 9 q^(3 + 22 n) - 441 q^(5 + 22 n) + 509 q^(7 + 22 n) - 9 q^(9 + 22 n) + 36 q^(4 + 24 n) - 560 q^(6 + 24 n) + 187 q^(8 + 24 n) + 84 q^(5 + 26 n) - 448 q^(7 + 26 n) + 35 q^(9 + 26 n) + 126 q^(6 + 28 n) - 231 q^(8 + 28 n) + 126 q^(7 + 30 n) - 63 q^(9 + 30 n) + 84 q^(8 + 32 n) - 7 q^(10 + 32 n) + 36 q^(9 + 34 n) + q^(11 + 34 n) + 9 q^(10 + 36 n) + q^(11 + 38 n)) QPochhammer[q, q^2, n]^16 QPochhammer[q^2, q^2, 4 n])/((1 + q^(4 n)) (1 + q^(1 + 2 n))^9 QPochhammer[q^2, q^2, 2 n]^10), {n, 0, Infinity}]

= QPochhammer[q, q^2]^8/QPochhammer[q^2, q^2]^8
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Example XIV:

Sum[((-1)^n q^ n^2 (1 + q^(2 n) + q^(4 n) + 5 q^(1 + 2 n) - 5 q^(1 + 4 n) + 10 q^(2 + 4 n) - 5 q^(1 + 6 n) - 16 q^(2 + 6 n) + 10 q^(3 + 6 n) - q^(1 + 8 n) - 11 q^(2 + 8 n) - 25 q^(3 + 8 n) + 5 q^(4 + 8 n) + 19 q^(2 + 10 n) - 14 q^(3 + 10 n) - 15 q^(4 + 10 n) + q^(5 + 10 n) + q^(2 + 12 n) + 46 q^(3 + 12 n) - 4 q^(4 + 12 n) - 4 q^(5 + 12 n) - 4 q^(2 + 14 n) - 4 q^(3 + 14 n) + 46 q^(4 + 14 n) + q^(5 + 14 n) + q^(2 + 16 n) - 15 q^(3 + 16 n) - 14 q^(4 + 16 n) + 19 q^(5 + 16 n) + 5 q^(3 + 18 n) - 25 q^(4 + 18 n) - 11 q^(5 + 18 n) - q^(6 + 18 n) + 10 q^(4 + 20 n) - 16 q^(5 + 20 n) - 5 q^(6 + 20 n) + 10 q^(5 + 22 n) - 5 q^(6 + 22 n) + q^(7 + 22 n) + 5 q^(6 + 24 n) + q^(7 + 24 n) + q^(7 + 26 n)) QPochhammer[q, q^2, n]^10 QPochhammer[q, q^2, 2 n] QPochhammer[q^2, q^2, 3 n])/((1 - q^n + q^(2 n)) (1 + q^n + q^(2 n)) (1 + q^(1 + 2 n))^5 QPochhammer[q^2, q^2, n]^5 QPochhammer[q^2, q^2, 2 n]^5), {n, 0, Infinity}]

= QPochhammer[q, q^2]^8/QPochhammer[q^2, q^2]^8
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Example XV:

Sum[(q^(2 b n + 3 d n + 2 n^2) (q^(4 b) - q^(4 + 2 b + 3 c + 8 n) - q^(1 + 5 b + 3 d + 8 n) - q^(1 + 5 b + 3 c + 3 d + 8 n) - q^(1 + 3 b + 3 c + 10 n) - q^(3 + 3 b + 3 c + 10 n) - q^(1 + 3 b + 3 c + 3 d + 10 n) - q^(3 + 3 b + 3 c + 3 d + 10 n) - q^(3 + b + 3 c + 12 n) - q^(3 + b + 6 c + 12 n) - q^(4 b + 3 c + 3 d + 12 n) + q^(2 + 4 b + 3 c + 3 d + 12 n) + q^(4 + 4 b + 3 c + 3 d + 12 n) + q^(4 + 2 b + 3 c + 14 n) + q^(4 + 2 b + 6 c + 14 n) + q^(4 + 2 b + 3 c + 3 d + 14 n) + q^(1 + 5 b + 3 c + 3 d + 14 n) + q^(4 + 2 b + 6 c + 3 d + 14 n) + q^(1 + 5 b + 3 c + 6 d + 14 n) + q^(4 + 6 c + 16 n) + q^(6 + 6 c + 16 n) + q^(1 + 3 b + 3 c + 3 d + 16 n) + q^(3 + 3 b + 3 c + 3 d + 16 n) + q^(1 + 3 b + 6 c + 3 d + 16 n) + q^(3 + 3 b + 6 c + 3 d + 16 n) + q^(3 + b + 6 c + 18 n) + q^(3 + b + 6 c + 3 d + 18 n) - q^(5 + 3 b + 6 c + 3 d + 22 n) - q^(5 + 3 b + 6 c + 6 d + 22 n) - q^(5 + b + 6 c + 3 d + 24 n) - q^(7 + b + 6 c + 3 d + 24 n) - q^(5 + b + 9 c + 3 d + 24 n) - q^(7 + b + 9 c + 3 d + 24 n) - q^(2 + 4 b + 6 c + 6 d + 24 n) - q^(4 + 4 b + 6 c + 6 d + 24 n) - q^(7 - b + 9 c + 26 n) - q^(4 + 2 b + 6 c + 3 d + 26 n) - q^(7 - b + 9 c + 3 d + 26 n) - q^(4 + 2 b + 9 c + 3 d + 26 n) - q^(4 + 2 b + 6 c + 6 d + 26 n) - q^(4 + 2 b + 9 c + 6 d + 26 n) - q^(4 + 9 c + 3 d + 28 n) - q^(6 + 9 c + 3 d + 28 n) + q^(8 + 9 c + 3 d + 28 n) + q^(5 + 3 b + 6 c + 6 d + 28 n) + q^(5 + 3 b + 9 c + 6 d + 28 n) + q^(5 + b + 9 c + 3 d + 30 n) + q^(7 + b + 9 c + 3 d + 30 n) + q^(5 + b + 9 c + 6 d + 30 n) + q^(7 + b + 9 c + 6 d + 30 n) + q^(7 - b + 9 c + 3 d + 32 n) + q^(7 - b + 12 c + 3 d + 32 n) + q^(4 + 2 b + 9 c + 6 d + 32 n) - q^(8 + 12 c + 6 d + 40 n)) QPochhammer[q^(1 - b), q^2, 2 n] QPochhammer[q^(1 + b), q^2, n] QPochhammer[q^(3 - 3 b + 3 c), q^6, n] QPochhammer[q^(1 - b + 3 c), q^2, 2 n] QPochhammer[q^(3 - 3 b + 3 c - 3 d), q^6, n] QPochhammer[q^(1 + b + 3 d), q^2, n] QPochhammer[q^(6 + 3 c + 3 d), q^6, 2 n])/(((1 - q^(3 c + 3 d + 12 n)) (q^(4 b) - q^(2 + 2 b + 3 c + 8 n) - q^(4 + 2 b + 3 c + 8 n) + q^(6 + 6 c + 16 n))) QPochhammer[q^6, q^6, n] QPochhammer[q^(6 + 3 c), q^6, n] QPochhammer[q^(2 - 2 b + 3 c), q^2, 4 n] QPochhammer[q^(6 + 3 d), q^6, n] QPochhammer[q^(2 + 2 b + 3 d), q^2, 2 n] QPochhammer[q^(6 + 3 c + 3 d), q^6, n]), {n, 0, Infinity}]

= (QPochhammer[q^(3 - 3 b + 3 c), q^6] QPochhammer[q^(1 - b + 3 c), q^2] QPochhammer[q^(1 + b + 3 d), q^2] QPochhammer[q^(3 + 3 b + 3 d), q^6])/(QPochhammer[q^(6 + 3 c), q^6] QPochhammer[q^(2 - 2 b + 3 c), q^2] QPochhammer[q^(6 + 3 d), q^6] QPochhammer[q^(2 + 2 b + 3 d), q^2])
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Example XVI:

Sum[(q^(2 b n + 2 c n + 2 n^2) (1 - q^(1 + b + 4 c + 8 n)) QPochhammer[q^(1 + b - 2 c), q^2, n] QPochhammer[q^(1 - b + 2 c), q^2, 2 n] QPochhammer[q^(1 + b + 4 c), q^2, n])/(QPochhammer[q^6, q^6, n] QPochhammer[q^(2 + 2 b + 2 c), q^2, 2 n] QPochhammer[q^(6 + 6 c), q^6, n]), {n, 0, Infinity}]

= (QPochhammer[q^(3 + 3 b), q^6] QPochhammer[q^(1 + b + 4 c), q^2])/(QPochhammer[q^(2 + 2 b + 2 c), q^2] QPochhammer[q^(6 + 6 c), q^6])
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Example XVII:

Sum[(q^(b n + c n + 2 d n + 2 n^2) (1 + q^(b + c + 2 d + 4 n) - q^(2 + b + c + 2 d + 4 n) - q^(1 + 2 b + c + d + 6 n) - q^(1 + b + 2 c + d + 6 n) - q^(1 + b + 3 d + 6 n) - q^(1 + c + 3 d + 6 n) - q^(1 + 3 b + 2 c + 3 d + 6 n) - q^(1 + 2 b + 3 c + 3 d + 6 n) - q^(1 + 2 b + c + 5 d + 6 n) - q^(1 + b + 2 c + 5 d + 6 n) + q^(2 + b + c + 2 d + 8 n) + q^(2 + 3 b + 3 c + 2 d + 8 n) + q^(2 + 3 b + c + 4 d + 8 n) + q^(2 + 2 b + 2 c + 4 d + 8 n) + q^(2 + b + 3 c + 4 d + 8 n) + q^(2 + b + c + 6 d + 8 n) + q^(2 + 3 b + 3 c + 6 d + 8 n) + q^(2 + 2 b + 2 c + 4 d + 12 n) + q^(2 + 4 b + 4 c + 4 d + 12 n) + q^(2 + 4 b + 2 c + 6 d + 12 n) + q^(2 + 3 b + 3 c + 6 d + 12 n) + q^(2 + 2 b + 4 c + 6 d + 12 n) + q^(2 + 2 b + 2 c + 8 d + 12 n) + q^(2 + 4 b + 4 c + 8 d + 12 n) - q^(3 + 4 b + 3 c + 5 d + 14 n) - q^(3 + 3 b + 4 c + 5 d + 14 n) - q^(3 + 3 b + 2 c + 7 d + 14 n) - q^(3 + 2 b + 3 c + 7 d + 14 n) - q^(3 + 5 b + 4 c + 7 d + 14 n) - q^(3 + 4 b + 5 c + 7 d + 14 n) - q^(3 + 4 b + 3 c + 9 d + 14 n) - q^(3 + 3 b + 4 c + 9 d + 14 n) - q^(2 + 4 b + 4 c + 8 d + 16 n) + q^(4 + 4 b + 4 c + 8 d + 16 n) + q^(4 + 5 b + 5 c + 10 d + 20 n)) QPochhammer[q^(1 + b - d), q^2, n] QPochhammer[q^(1 + c - d), q^2, n] QPochhammer[q^(1 - b + d), q^2, n] QPochhammer[q^(1 - c + d), q^2, n] QPochhammer[q^(1 + 2 b + c + d), q^2, n] QPochhammer[q^(1 + b + 2 c + d), q^2, n] QPochhammer[q^(1 + b + 3 d), q^2, n] QPochhammer[q^(1 + c + 3 d), q^2, n] QPochhammer[q^(4 + 2 b + 2 c + 4 d), q^4, 2 n])/((1 - q^(2 + b + c + 2 d + 4 n)) (1 + q^(b + c + 2 d + 4 n)) QPochhammer[q^4, q^4, n] QPochhammer[q^(4 + 2 b + 2 c), q^4, n] QPochhammer[q^(4 + 2 b + 2 d), q^4, n] QPochhammer[q^(2 + b + c + 2 d), q^2, 2 n]^2 QPochhammer[q^(4 + 2 c + 2 d), q^4, n] QPochhammer[q^(4 + 4 d), q^4, n] QPochhammer[q^(4 + 2 b + 2 c + 4 d), q^4, n]), {n, 0, Infinity}]

= (QPochhammer[q^(2 + 2 b + 2 c), q^4] QPochhammer[q^(2 + 4 c), q^4] QPochhammer[q^(4 + 4 c), q^4] QPochhammer[q^(1 + 2 b + c + d), q^2] QPochhammer[q^(1 + b + 2 c + d), q^2] QPochhammer[q^(1 + b + 3 d), q^2] QPochhammer[q^(1 + c + 3 d), q^2])/(QPochhammer[q^(2 + 2 b + 2 c), q^2] QPochhammer[q^(2 + 4 c), q^2] QPochhammer[q^(4 + 2 b + 2 d), q^4] QPochhammer[q^(2 + b + c + 2 d), q^2]^2 QPochhammer[q^(4 + 2 c + 2 d), q^4] QPochhammer[q^(4 + 4 d), q^4])
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Example XVIII:

Sum[(q^(b n + c n + 2 d n + 6 n^2) (1 + q^(b + c + 2 d + 12 n) - q^(6 + b + c + 2 d + 12 n) - q^(3 + 2 b + c + d + 18 n) - q^(1 + b + 2 c + d + 18 n) - q^(5 + b + 3 d + 18 n) - q^(3 + c + 3 d + 18 n) - q^(3 + 3 b + 2 c + 3 d + 18 n) - q^(1 + 2 b + 3 c + 3 d + 18 n) - q^(5 + 2 b + c + 5 d + 18 n) - q^(3 + b + 2 c + 5 d + 18 n) + q^(6 + b + c + 2 d + 24 n) + q^(4 + 3 b + 3 c + 2 d + 24 n) + q^(8 + 3 b + c + 4 d + 24 n) + q^(6 + 2 b + 2 c + 4 d + 24 n) + q^(4 + b + 3 c + 4 d + 24 n) + q^(8 + b + c + 6 d + 24 n) + q^(6 + 3 b + 3 c + 6 d + 24 n) + q^(6 + 2 b + 2 c + 4 d + 36 n) + q^(4 + 4 b + 4 c + 4 d + 36 n) + q^(8 + 4 b + 2 c + 6 d + 36 n) + q^(6 + 3 b + 3 c + 6 d + 36 n) + q^(4 + 2 b + 4 c + 6 d + 36 n) + q^(8 + 2 b + 2 c + 8 d + 36 n) + q^(6 + 4 b + 4 c + 8 d + 36 n) - q^(9 + 4 b + 3 c + 5 d + 42 n) - q^(7 + 3 b + 4 c + 5 d + 42 n) - q^(11 + 3 b + 2 c + 7 d + 42 n) - q^(9 + 2 b + 3 c + 7 d + 42 n) - q^(9 + 5 b + 4 c + 7 d + 42 n) - q^(7 + 4 b + 5 c + 7 d + 42 n) - q^(11 + 4 b + 3 c + 9 d + 42 n) - q^(9 + 3 b + 4 c + 9 d + 42 n) - q^(6 + 4 b + 4 c + 8 d + 48 n) + q^(12 + 4 b + 4 c + 8 d + 48 n) + q^(12 + 5 b + 5 c + 10 d + 60 n)) QPochhammer[q^(3 + b - d), q^6, n] QPochhammer[q^(1 + c - d), q^6, n] QPochhammer[q^(3 - b + d), q^6, n] QPochhammer[q^(5 - c + d), q^6, n] QPochhammer[q^(3 + 2 b + c + d), q^6, n] QPochhammer[q^(1 + b + 2 c + d), q^6, n] QPochhammer[q^(5 + b + 3 d), q^6, n] QPochhammer[q^(3 + c + 3 d), q^6, n] QPochhammer[q^(12 + 2 b + 2 c + 4 d), q^12, 2 n])/((1 - q^(2 + 2 b + 2 d + 12 n)) (1 + q^(b + c + 2 d + 12 n)) (1 - q^(6 + b + c + 2 d + 12 n)) (1 - q^(2 + 4 d + 12 n)) QPochhammer[q^12, q^12, n] QPochhammer[q^(10 + 2 b + 2 c), q^12, n] QPochhammer[q^(2 + 2 b + 2 d), q^12, n] QPochhammer[q^(6 + b + c + 2 d), q^6, 2 n]^2 QPochhammer[q^(10 + 2 c + 2 d), q^12, n] QPochhammer[q^(2 + 4 d), q^12, n] QPochhammer[q^(12 + 2 b + 2 c + 4 d), q^12, n]), {n, 0, Infinity}]

= (QPochhammer[q^(4 + 2 b + 2 c), q^12] QPochhammer[q^(6 + 4 c), q^12] QPochhammer[q^(12 + 4 c), q^12] QPochhammer[q^(3 + 2 b + c + d), q^6] QPochhammer[q^(1 + b + 2 c + d), q^6] QPochhammer[q^(5 + b + 3 d), q^6] QPochhammer[q^(3 + c + 3 d), q^6])/(QPochhammer[q^(4 + 2 b + 2 c), q^6] QPochhammer[q^(6 + 4 c), q^6] QPochhammer[q^(2 + 2 b + 2 d), q^12] QPochhammer[q^(6 + b + c + 2 d), q^6]^2 QPochhammer[q^(10 + 2 c + 2 d), q^12] QPochhammer[q^(2 + 4 d), q^12])
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Example XIX:

Sum[(q^(2 n) (q^b + q^(2 + b + c + 2 d + 6 n) + q^d (q^c - q^(2 n) (q^(1 + b) + q^c + q^(1 + b + c)))) QPochhammer[q^(1 + b), q^2, n] QPochhammer[q^(1 + b - c), q^2, n] QPochhammer[q^(1 + d), q^2, n] QPochhammer[q^(1 + c + d), q^2, n])/((1 - q^(2 + b + d + 4 n)) QPochhammer[q^2, q^2, n] QPochhammer[q^(2 + b + d), q^2, 2 n] QPochhammer[q^(2 - b + c + d), q^2, n]), {n, 0, Infinity}]

= (q^(-1 + c) QPochhammer[q^(1 + b), q^2] QPochhammer[q^(1 + b - c), q^2] QPochhammer[q^(1 + d), q^2] QPochhammer[q^(1 + c + d), q^2])/(QPochhammer[q^2, q^2] QPochhammer[q^(2 + b + d), q^2] QPochhammer[q^(2 - b + c + d), q^2]) Sum[q^(n + b n - c n)/QPochhammer[q^(1 + b), q^2, n], {n, 1, Infinity}]
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Example XX:

Sum[(q^(2 n) (q^(3 d) + q^(a + b + 2 d) + q^(a + b + 3 d + e) - q^(1 + a + 2 d + 2 n) - q^(1 + b + 2 d + 2 n) - q^(3 d + 2 n) - q^(1 + 2 a + b + 2 d + e + 2 n) - q^(1 + a + 2 b + 2 d + e + 2 n) - q^(1 + a + 3 d + e + 2 n) - q^(1 + b + 3 d + e + 2 n) - q^(a + b + 3 d + e + 2 n) - q^(4 d + e + 2 n) + q^(2 + 2 a + 2 d + e + 4 n) - q^(a + b + 2 d + e + 4 n) + q^(2 + 2 b + 2 d + e + 4 n) + q^(1 + a + 3 d + e + 4 n) + q^(1 + b + 3 d + e + 4 n) + q^(4 d + e + 4 n) + q^(2 + 2 d + 6 n) + q^(2 + a + b + 2 d + e + 6 n) + q^(2 + 3 d + e + 6 n) + q^(2 + 2 a + 2 b + 2 d + 2 e + 6 n) + q^(2 + a + b + 3 d + 2 e + 6 n) + q^(2 + 4 d + 2 e + 6 n) + q^(2 + a + b + d + e + 8 n) + q^(4 + a + b + d + e + 8 n) + q^(1 + a + 2 d + e + 8 n) + q^(1 + b + 2 d + e + 8 n) - q^(2 + 3 d + e + 8 n) + q^(2 + 2 a + 2 b + d + 2 e + 8 n) + q^(4 + 2 a + 2 b + d + 2 e + 8 n) + q^(2 + a + b + 2 d + 2 e + 8 n) + q^(4 + a + b + 2 d + 2 e + 8 n) + q^(1 + 2 a + b + 2 d + 2 e + 8 n) + q^(1 + a + 2 b + 2 d + 2 e + 8 n) + q^(1 + a + 3 d + 2 e + 8 n) + q^(1 + b + 3 d + 2 e + 8 n) - q^(2 + a + b + 3 d + 2 e + 8 n) - q^(2 + 4 d + 2 e + 8 n) - q^(2 + a + b + d + e + 10 n) - q^(4 + a + b + d + e + 10 n) - q^(2 + 2 d + e + 10 n) - q^(4 + 2 d + e + 10 n) - q^(5 + 2 a + b + d + 2 e + 10 n) - q^(5 + a + 2 b + d + 2 e + 10 n) - q^(2 + 2 a + 2 b + d + 2 e + 10 n) - q^(4 + 2 a + 2 b + d + 2 e + 10 n) - q^(2 + 2 a + 2 d + 2 e + 10 n) - q^(4 + 2 a + 2 d + 2 e + 10 n) - 3 q^(2 + a + b + 2 d + 2 e + 10 n) - 3 q^(4 + a + b + 2 d + 2 e + 10 n) - q^(2 + 2 b + 2 d + 2 e + 10 n) - q^(4 + 2 b + 2 d + 2 e + 10 n) - q^(2 + 3 d + 2 e + 10 n) - q^(4 + 3 d + 2 e + 10 n) - q^(1 + a + 3 d + 2 e + 10 n) - q^(1 + b + 3 d + 2 e + 10 n) - q^(2 + 2 a + 2 b + 2 d + 3 e + 10 n) - q^(4 + 2 a + 2 b + 2 d + 3 e + 10 n) - q^(2 + a + b + 3 d + 3 e + 10 n) - q^(4 + a + b + 3 d + 3 e + 10 n) - q^(4 + 2 a + 2 b + 2 e + 12 n) - q^(4 + a + b + d + 2 e + 12 n) + q^(5 + 2 a + b + d + 2 e + 12 n) + q^(5 + a + 2 b + d + 2 e + 12 n) + q^(5 + a + 2 d + 2 e + 12 n) + q^(5 + b + 2 d + 2 e + 12 n) + q^(2 + a + b + 2 d + 2 e + 12 n) + q^(4 + a + b + 2 d + 2 e + 12 n) + q^(2 + 3 d + 2 e + 12 n) + q^(4 + 3 d + 2 e + 12 n) - q^(4 + 2 a + 2 b + d + 3 e + 12 n) + q^(5 + 2 a + b + 2 d + 3 e + 12 n) + q^(5 + a + 2 b + 2 d + 3 e + 12 n) + q^(2 + a + b + 3 d + 3 e + 12 n) + q^(4 + a + b + 3 d + 3 e + 12 n) + q^(4 + 2 a + 2 b + 2 e + 14 n) + q^(4 + a + b + d + 2 e + 14 n) + q^(4 + 2 d + 2 e + 14 n) + q^(4 + 2 a + 2 b + d + 3 e + 14 n) + q^(4 + a + b + 2 d + 3 e + 14 n) + q^(4 + 2 a + 2 b + 2 d + 4 e + 14 n) + q^(6 + 2 a + 2 b + 3 e + 16 n) + q^(5 + 2 a + b + d + 3 e + 16 n) + q^(5 + a + 2 b + d + 3 e + 16 n) + q^(4 + 2 a + 2 d + 3 e + 16 n) - q^(6 + a + b + 2 d + 3 e + 16 n) + q^(4 + 2 b + 2 d + 3 e + 16 n) - q^(6 + 2 a + 2 b + 3 e + 18 n) - q^(6 + a + b + d + 3 e + 18 n) - q^(5 + 2 a + b + d + 3 e + 18 n) - q^(5 + a + 2 b + d + 3 e + 18 n) - q^(5 + a + 2 d + 3 e + 18 n) - q^(5 + b + 2 d + 3 e + 18 n) - q^(6 + 2 a + 2 b + d + 4 e + 18 n) - q^(5 + 2 a + b + 2 d + 4 e + 18 n) - q^(5 + a + 2 b + 2 d + 4 e + 18 n) + q^(6 + a + b + d + 3 e + 20 n) + q^(6 + 2 a + 2 b + d + 4 e + 20 n) + q^(6 + a + b + 2 d + 4 e + 20 n)) QPochhammer[q^(1 - a), q^2, n] QPochhammer[q^(1 - b), q^2, n] QPochhammer[q^(1 + a - d), q^2, n] QPochhammer[q^(1 + b - d), q^2, n] QPochhammer[q^(2 + e), q^2, 3 n] QPochhammer[q^(1 + a + e), q^2, n] QPochhammer[q^(1 + b + e), q^2, n])/((-q^d + q^(2 + 4 n)) (-1 + q^(2 + e + 4 n)) (-q^d + q^(2 + a + b + e + 4 n)) (-1 + q^(e + 6 n)) QPochhammer[q^2, q^2, n] QPochhammer[q^(2 - d), q^2, 2 n] QPochhammer[q^(2 - a - b + d), q^2, n] QPochhammer[q^(2 + e), q^2, 2 n] QPochhammer[q^(2 + a + b - d + e), q^2, 2 n] QPochhammer[q^(2 + d + e), q^2, n]), {n, 0, Infinity}]

= (q^(-1 + a + d) QPochhammer[q^(1 - a), q^2] QPochhammer[q^(1 - b), q^2] QPochhammer[q^(1 + a - d), q^2] QPochhammer[q^(1 + b - d), q^2] QPochhammer[q^(1 + a + e), q^2] QPochhammer[q^(1 + b + e), q^2])/(QPochhammer[q^2, q^2] QPochhammer[q^(2 - d), q^2] QPochhammer[q^(2 - a - b + d), q^2] QPochhammer[q^(2 + a + b - d + e), q^2] QPochhammer[q^(2 + d + e), q^2])*Sum[q^(n + b n - d n)/QPochhammer[q^(1 + a - d), q^2, n], {n, 1, Infinity}]
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Example XXI:

Sum[(q^(2 n) (q^(2 b + 2 d) + q^(c + 2 d) + q^(2 b + c + 4 d) - q^(1 + d + 2 n) - q^(1 + 2 b + d + 2 n) - q^(1 + 2 b + 3 d + 2 n) - q^(1 + 4 b + 3 d + 2 n) - q^(1 + c + 3 d + 2 n) - q^(1 + 2 b + c + 3 d + 2 n) + q^(2 + 2 d + 4 n) - q^(2 b + 2 d + 4 n) + q^(2 + 4 b + 2 d + 4 n) + q^(2 + 2 b - c + 8 n) + q^(4 + 2 b - c + 8 n) + q^(2 + 2 b + 2 d + 8 n) + q^(4 + 2 b + 2 d + 8 n) + q^(2 + 4 b - c + 2 d + 8 n) + q^(4 + 4 b - c + 2 d + 8 n) - q^(5 + 2 b - c + d + 10 n) - q^(5 + 4 b - c + d + 10 n) - q^(4 + 4 b - 2 c + 12 n) - q^(4 + 2 b - c + 12 n) - q^(4 + 4 b - c + 2 d + 12 n) + q^(6 + 4 b - 2 c + 16 n)) QPochhammer[q^(1 - d), q^2, n] QPochhammer[q^(1 - 2 b - d), q^2, n] QPochhammer[q^(1 - c - d), q^2, n] QPochhammer[q^(1 + 2 b - c - d), q^2, n] QPochhammer[q^(1 + d), q^2, n] QPochhammer[q^(1 + 2 b + d), q^2, n])/((1 - q^(2 + 4 n)) (q^(c + 2 d) - q^(2 + 4 n)) (1 - q^(2 + 2 b - c + 4 n)) QPochhammer[q^2, q^2, 2 n] QPochhammer[q^(2 + 2 b - c), q^2, 2 n] QPochhammer[q^(2 - c - 2 d), q^2, 2 n]), {n, 0, Infinity}]

= -(q^(-1 + d) QPochhammer[q^(1 - d), q^2] QPochhammer[q^(1 - c - d), q^2] QPochhammer[q^(1 + 2 b - c - d), q^2] QPochhammer[q^(1 + d), q^2])/(QPochhammer[q^2, q^2]^2 QPochhammer[q^(2 + 2 b - c), q^2] QPochhammer[q^(2 - c - 2 d), q^2])*Sum[((-1)^n q^(2 b n + d n + n^2))/(1 - q^(-1 + d + 2 n)), {n, -Infinity, Infinity}]