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1: 16.18 Special Cases
16.18.1 F q p ( a 1 , , a p b 1 , , b q ; z ) = ( k = 1 q Γ ( b k ) / k = 1 p Γ ( a k ) ) G p , q + 1 1 , p ( - z ; 1 - a 1 , , 1 - a p 0 , 1 - b 1 , , 1 - b q ) = ( k = 1 q Γ ( b k ) / k = 1 p Γ ( a k ) ) G q + 1 , p p , 1 ( - 1 z ; 1 , b 1 , , b q a 1 , , a p ) .
2: 26.12 Plane Partitions
26.12.21 π B ( r , s , t ) q | π | = ( h , j , k ) B ( r , s , t ) 1 - q h + j + k - 1 1 - q h + j + k - 2 = h = 1 r j = 1 s 1 - q h + j + t - 1 1 - q h + j - 1 ,
26.12.22 π B ( r , r , t ) π  symmetric q | π | = h = 1 r 1 - q 2 h + t - 1 1 - q 2 h - 1 1 h < j r 1 - q 2 ( h + j + t - 1 ) 1 - q 2 ( h + j - 1 ) .
26.12.23 π B ( r , r , r ) π  cyclically symmetric q | π | = h = 1 r 1 - q 3 h - 1 1 - q 3 h - 2 1 h < j r 1 - q 3 ( h + 2 j - 1 ) 1 - q 3 ( h + j - 1 ) = h = 1 r ( 1 - q 3 h - 1 1 - q 3 h - 2 j = h r 1 - q 3 ( r + h + j - 1 ) 1 - q 3 ( 2 h + j - 1 ) ) .
3: 20.4 Values at z = 0
20.4.2 θ 1 ( 0 , q ) = 2 q 1 / 4 n = 1 ( 1 - q 2 n ) 3 = 2 q 1 / 4 ( q 2 ; q 2 ) 3 ,
20.4.3 θ 2 ( 0 , q ) = 2 q 1 / 4 n = 1 ( 1 - q 2 n ) ( 1 + q 2 n ) 2 ,
20.4.4 θ 3 ( 0 , q ) = n = 1 ( 1 - q 2 n ) ( 1 + q 2 n - 1 ) 2 ,
20.4.5 θ 4 ( 0 , q ) = n = 1 ( 1 - q 2 n ) ( 1 - q 2 n - 1 ) 2 .
4: 16.17 Definition
16.17.1 G p , q m , n ( z ; a ; b ) = G p , q m , n ( z ; a 1 , , a p b 1 , , b q ) = 1 2 π i L ( = 1 m Γ ( b - s ) = 1 n Γ ( 1 - a + s ) / ( = m q - 1 Γ ( 1 - b + 1 + s ) = n p - 1 Γ ( a + 1 - s ) ) ) z s d s ,
16.17.3 A p , q , k m , n ( z ) = = 1 k m Γ ( b - b k ) = 1 n Γ ( 1 + b k - a ) z b k / ( = m q - 1 Γ ( 1 + b k - b + 1 ) = n p - 1 Γ ( a + 1 - b k ) ) .
5: 20.5 Infinite Products and Related Results
20.5.1 θ 1 ( z , q ) = 2 q 1 / 4 sin z n = 1 ( 1 - q 2 n ) ( 1 - 2 q 2 n cos ( 2 z ) + q 4 n ) ,
20.5.2 θ 2 ( z , q ) = 2 q 1 / 4 cos z n = 1 ( 1 - q 2 n ) ( 1 + 2 q 2 n cos ( 2 z ) + q 4 n ) ,
20.5.3 θ 3 ( z , q ) = n = 1 ( 1 - q 2 n ) ( 1 + 2 q 2 n - 1 cos ( 2 z ) + q 4 n - 2 ) ,
20.5.4 θ 4 ( z , q ) = n = 1 ( 1 - q 2 n ) ( 1 - 2 q 2 n - 1 cos ( 2 z ) + q 4 n - 2 ) .
20.5.9 θ 3 ( π z | τ ) = n = - p 2 n q n 2 = n = 1 ( 1 - q 2 n ) ( 1 + q 2 n - 1 p 2 ) ( 1 + q 2 n - 1 p - 2 ) ,
6: 26.16 Multiset Permutations
26.16.1 [ a 1 + a 2 + + a n a 1 , a 2 , , a n ] q = k = 1 n - 1 [ a k + a k + 1 + + a n a k ] q ,
7: 16.11 Asymptotic Expansions
16.11.2 H p , q ( z ) = m = 1 p k = 0 ( - 1 ) k k ! Γ ( a m + k ) ( = 1 m p Γ ( a - a m - k ) / = 1 q Γ ( b - a m - k ) ) z - a m - k .
8: 17.2 Calculus
17.2.4 ( a ; q ) = j = 0 ( 1 - a q j ) ,
17.2.49 1 + n = 1 q n 2 ( 1 - q ) ( 1 - q 2 ) ( 1 - q n ) = n = 0 1 ( 1 - q 5 n + 1 ) ( 1 - q 5 n + 4 ) ,
17.2.50 1 + n = 1 q n 2 + n ( 1 - q ) ( 1 - q 2 ) ( 1 - q n ) = n = 0 1 ( 1 - q 5 n + 2 ) ( 1 - q 5 n + 3 ) .
9: 23.17 Elementary Properties
23.17.8 η ( τ ) = q 1 / 12 n = 1 ( 1 - q 2 n ) ,
10: 5.18 q -Gamma and q -Beta Functions
5.18.3 ( a ; q ) = k = 0 ( 1 - a q k ) .