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1: 17.2 Calculus
17.2.6_2 ( - q ; q ) = 1 2 exp ( π 2 12 t + t 24 ) ( q ^ 1 2 ; q ^ ) , t > 0 .
17.2.7 ( a ; q - 1 ) n = ( a - 1 ; q ) n ( - a ) n q - ( n 2 ) ,
17.2.8 ( a ; q - 1 ) n ( b ; q - 1 ) n = ( a - 1 ; q ) n ( b - 1 ; q ) n ( a b ) n ,
17.2.11 ( a q - n ; q ) n = ( q / a ; q ) n ( - a q ) n q - ( n 2 ) ,
17.2.12 ( a q - n ; q ) n ( b q - n ; q ) n = ( q / a ; q ) n ( q / b ; q ) n ( a b ) n .
2: 17.14 Constant Term Identities
17.14.2 n = 0 q n ( n + 1 ) ( q 2 ; q 2 ) n ( - q ; q 2 ) n + 1 =  coeff. of  z 0  in  ( - z q ; q 2 ) ( - z - 1 q ; q 2 ) ( q 2 ; q 2 ) ( z - 1 q 2 ; q 2 ) ( - q ; q 2 ) ( z - 1 q ; q 2 ) = 1 ( - q ; q 2 )  coeff. of  z 0  in  ( - z q ; q 2 ) ( - z - 1 q ; q 2 ) ( q 2 ; q 2 ) ( z - 1 q ; q ) = H ( q ) ( - q ; q 2 ) ,
17.14.3 n = 0 q n ( n + 1 ) ( q 2 ; q 2 ) n ( - q ; q 2 ) n + 1 =  coeff. of  z 0  in  ( - z q ; q 2 ) ( - z - 1 q ; q 2 ) ( q 2 ; q 2 ) ( z - 1 ; q 2 ) ( - q ; q 2 ) ( z - 1 q ; q 2 ) = 1 ( - q ; q 2 )  coeff. of  z 0  in  ( - z q ; q 2 ) ( - z - 1 q ; q 2 ) ( q 2 ; q 2 ) ( z - 1 ; q ) = G ( q ) ( - q ; q 2 ) ,
17.14.4 n = 0 q n 2 ( q 2 ; q 2 ) n ( q ; q 2 ) n =  coeff. of  z 0  in  ( - z q ; q 2 ) ( - z - 1 q ; q 2 ) ( q 2 ; q 2 ) ( - z - 1 ; q 2 ) ( q ; q 2 ) ( z - 1 ; q 2 ) = 1 ( q ; q 2 )  coeff. of  z 0  in  ( - z q ; q 2 ) ( - z - 1 q ; q 2 ) ( q 2 ; q 2 ) ( z - 2 ; q 4 ) = G ( q 4 ) ( q ; q 2 ) ,
17.14.5 n = 0 q n 2 + 2 n ( q 2 ; q 2 ) n ( q ; q 2 ) n + 1 =  coeff. of  z 0  in  ( - z q ; q 2 ) ( - z - 1 q ; q 2 ) ( q 2 ; q 2 ) ( - q 2 z - 1 ; q 2 ) ( q ; q 2 ) ( z - 1 q 2 ; q 2 ) = 1 ( q ; q 2 )  coeff. of  z 0  in  ( - z q ; q 2 ) ( - z - 1 q ; q 2 ) ( q 2 ; q 2 ) ( q 4 z - 2 ; q 4 ) = H ( q 4 ) ( q ; q 2 ) .
3: 17.13 Integrals
17.13.1 - c d ( - q x / c ; q ) ( q x / d ; q ) ( - a x / c ; q ) ( b x / d ; q ) d q x = ( 1 - q ) ( q ; q ) ( a b ; q ) c d ( - c / d ; q ) ( - d / c ; q ) ( a ; q ) ( b ; q ) ( c + d ) ( - b c / d ; q ) ( - a d / c ; q ) ,
17.13.2 - c d ( - q x / c ; q ) ( q x / d ; q ) ( - x q α / c ; q ) ( x q β / d ; q ) d q x = Γ q ( α ) Γ q ( β ) Γ q ( α + β ) c d c + d ( - c / d ; q ) ( - d / c ; q ) ( - q β c / d ; q ) ( - q α d / c ; q ) .
17.13.3 0 t α - 1 ( - t q α + β ; q ) ( - t ; q ) d t = Γ ( α ) Γ ( 1 - α ) Γ q ( β ) Γ q ( 1 - α ) Γ q ( α + β ) ,
17.13.4 0 t α - 1 ( - c t q α + β ; q ) ( - c t ; q ) d q t = Γ q ( α ) Γ q ( β ) ( - c q α ; q ) ( - q 1 - α / c ; q ) Γ q ( α + β ) ( - c ; q ) ( - q / c ; q ) .
4: 18.28 Askey–Wilson Class
18.28.1 p n ( cos θ ) = p n ( cos θ ; a , b , c , d | q ) = a - n = 0 n q ( a b q , a c q , a d q ; q ) n - ( q - n , a b c d q n - 1 ; q ) ( q ; q ) j = 0 - 1 ( 1 - 2 a q j cos θ + a 2 q 2 j ) = a - n ( a b , a c , a d ; q ) n ϕ 3 4 ( q - n , a b c d q n - 1 , a e i θ , a e - i θ a b , a c , a d ; q , q ) .
18.28.5 h n = h 0 ( 1 - a b c d q n - 1 ) ( q , a b , a c , a d , b c , b d , c d ; q ) n ( 1 - a b c d q 2 n - 1 ) ( a b c d ; q ) n , n = 1 , 2 , .
18.28.9 Q n ( 1 2 ( a q - y + a - 1 q y ) ; a , b | q - 1 ) = ( - 1 ) n b n q - 1 2 n ( n - 1 ) ( ( a b ) - 1 ; q ) n ϕ 1 3 ( q - n , q - y , a - 2 q y ( a b ) - 1 ; q , q n a b - 1 ) .
18.28.10 y = 0 ( 1 - q 2 y a - 2 ) ( a - 2 , ( a b ) - 1 ; q ) y ( 1 - a - 2 ) ( q , b q a - 1 ; q ) y ( b a - 1 ) y q y 2 Q n ( 1 2 ( a q - y + a - 1 q y ) ; a , b | q - 1 ) Q m ( 1 2 ( a q - y + a - 1 q y ) ; a , b | q - 1 ) = ( q a - 2 ; q ) ( b a - 1 q ; q ) ( q , ( a b ) - 1 ; q ) n ( a b ) n q - n 2 δ n , m .
18.28.19 R n ( x ) = R n ( x ; α , β , γ , δ | q ) = = 0 n q ( q - n , α β q n + 1 ; q ) ( α q , β δ q , γ q , q ; q ) j = 0 - 1 ( 1 - q j x + γ δ q 2 j + 1 ) = ϕ 3 4 ( q - n , α β q n + 1 , q - y , γ δ q y + 1 α q , β δ q , γ q ; q , q ) , α q , β δ q , or γ q = q - N ; n = 0 , 1 , , N .
5: 18.1 Notation
q -Pochhammer Symbol
Infinite q -Product
q -Hahn Class OP’s
  • Little q -Jacobi: p n ( x ; a , b ; q ) .

  • Continuous q - 1 -Hermite: h n ( x | q )

  • 6: 5.18 q -Gamma and q -Beta Functions
    5.18.2 n ! q = 1 ( 1 + q ) ( 1 + q + + q n - 1 ) = ( q ; q ) n ( 1 - q ) - n .
    5.18.4 Γ q ( z ) = ( q ; q ) ( 1 - q ) 1 - z / ( q z ; q ) ,
    5.18.12 B q ( a , b ) = 0 1 t a - 1 ( t q ; q ) ( t q b ; q ) d q t , 0 < q < 1 , a > 0 , b > 0 .
    7: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
    18.29.2 Q n ( z ; a , b , c , d q ) z n ( a z - 1 , b z - 1 , c z - 1 , d z - 1 ; q ) ( z - 2 , b c , b d , c d ; q ) , n ; z , a , b , c , d , q fixed.
    8: 17.12 Bailey Pairs
    17.12.3 β n = j = 0 n α j ( q ; q ) n - j ( a q ; q ) n + j .
    9: 17.3 q -Elementary and q -Special Functions
    17.3.1 e q ( x ) = n = 0 ( 1 - q ) n x n ( q ; q ) n = 1 ( ( 1 - q ) x ; q ) ,
    17.3.2 E q ( x ) = n = 0 ( 1 - q ) n q ( n 2 ) x n ( q ; q ) n = ( - ( 1 - q ) x ; q ) .
    17.3.5 cos q ( x ) = 1 2 ( e q ( i x ) + e q ( - i x ) ) = n = 0 ( 1 - q ) 2 n ( - 1 ) n x 2 n ( q ; q ) 2 n ,
    17.3.6 Cos q ( x ) = 1 2 ( E q ( i x ) + E q ( - i x ) ) = n = 0 ( 1 - q ) 2 n q n ( 2 n - 1 ) ( - 1 ) n x 2 n ( q ; q ) 2 n .
    17.3.9 a m , s ( q ) = q - ( s 2 ) ( 1 - q ) s ( q ; q ) s j = 0 s ( - 1 ) j q ( j 2 ) [ s j ] q ( 1 - q s - j ) m ( 1 - q ) m .
    10: 17.7 Special Cases of Higher ϕ s r Functions
    17.7.2 ϕ 2 2 ( a 2 , b 2 a b q 1 2 , - a b q 1 2 ; q , - q ) = ( a 2 q , b 2 q ; q 2 ) ( q , a 2 b 2 q ; q 2 ) .
    17.7.20 k = 0 n 1 - a p k q k 1 - a ( a ; p ) k ( c ; q ) k ( q ; q ) k ( a p / c ; p ) k c - k = ( a p ; p ) n ( c q ; q ) n ( q ; q ) n ( a p / c ; p ) n c - n .
    17.7.21 k = 0 n ( 1 - a p k q k ) ( 1 - b p k q - k ) ( 1 - a ) ( 1 - b ) ( a , b ; p ) k ( c , a / ( b c ) ; q ) k ( q , a q / b ; q ) k ( a p / c , b c p ; p ) k q k = ( a p , b p ; p ) n ( c q , a q / ( b c ) ; q ) n ( q , a q / b ; q ) n ( a p / c , b c p ; p ) n ,
    17.7.23 ( 1 - a q ) ( 1 - b q ) k = 0 n ( a p k , b p - k ; q ) n - 1 ( 1 - ( a p 2 k / b ) ) ( p ; p ) n ( p ; p ) n - k ( a p k / b ; q ) n + 1 ( - 1 ) k p ( k 2 ) = δ n , 0 .