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11: 18.27 q -Hahn Class
18.27.4 y = 0 N Q n ( q - y ) Q m ( q - y ) ( α q , q - N ; q ) y ( α β q ) - y ( q , β - 1 q - N ; q ) y = h n δ n , m , n , m = 0 , 1 , , N .
18.27.9 v x = ( a - 1 x , c - 1 x ; q ) ( x , b c - 1 x ; q ) , 0 < a < q - 1 , 0 < b < q - 1 , c < 0 ,
18.27.12 v x = ( q x / c , - q x / d ; q ) ( q α + 1 x / c , - q β + 1 x / d ; q ) , α , β > - 1 , c , d > 0 .
18.27.22 = 0 ( h n ( q ; q ) h m ( q ; q ) + h n ( - q ; q ) h m ( - q ; q ) ) ( q + 1 , - q + 1 ; q ) q = ( q ; q ) n ( q , - 1 , - q ; q ) q n ( n - 1 ) / 2 δ n , m .
12: 17.9 Further Transformations of ϕ r r + 1 Functions
17.9.3 ϕ 1 2 ( a , b c ; q , z ) = ( a b z / c ; q ) ( b z / c ; q ) ϕ 2 3 ( a , c / b , 0 c , c q / ( b z ) ; q , q ) + ( a , b z , c / b ; q ) ( c , z , c / ( b z ) ; q ) ϕ 2 3 ( z , a b z / c , 0 b z , b z q / c ; q , q ) ,
17.9.8 ϕ 2 3 ( q - n , b , c d , e ; q , q ) = ( d e / ( b c ) ; q ) n ( e ; q ) n ( b c d ) n ϕ 2 3 ( q - n , d / b , d / c d , d e / ( b c ) ; q , q ) ,
17.9.9 ϕ 2 3 ( q - n , b , c d , e ; q , q ) = ( e / c ; q ) n ( e ; q ) n c n ϕ 2 3 ( q - n , c , d / b d , c q 1 - n / e ; q , b q e ) ,
13: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
18.29.1 ( b c , b d , c d ; q ) n ( Q n ( e i θ ; a , b , c , d q ) + Q n ( e - i θ ; a , b , c , d q ) ) ,
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.
14: 17.14 Constant Term Identities
17.14.1 ( q ; q ) a 1 + a 2 + + a n ( q ; q ) a 1 ( q ; q ) a 2 ( q ; q ) a n =  coeff. of  x 1 0 x 2 0 x n 0  in  1 j < k n ( x j x k ; q ) a j ( q x k x j ; q ) a k .
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 ) .
15: 5.18 q -Gamma and q -Beta Functions
5.18.1 ( a ; q ) n = k = 0 n - 1 ( 1 - a q k ) , n = 0 , 1 , 2 , ,
5.18.2 n ! q = 1 ( 1 + q ) ( 1 + q + + q n - 1 ) = ( q ; q ) n ( 1 - q ) - n .
5.18.3 ( a ; q ) = k = 0 ( 1 - a q k ) .
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 .
16: 17.6 ϕ 1 2 Function
17.6.14 n = 0 ( a ; q ) n ( b ; q 2 ) n z n ( q ; q ) n ( a z b ; q 2 ) n = ( a z , b z ; q 2 ) ( z , a z b ; q 2 ) ϕ 1 2 ( a , b b z ; q 2 , z q ) .
17: 17.4 Basic Hypergeometric Functions
17.4.1 ϕ s r + 1 ( a 0 , a 1 , a 2 , , a r b 1 , b 2 , , b s ; q , z ) = ϕ s r + 1 ( a 0 , a 1 , , a r ; b 1 , b 2 , , b s ; q , z ) = n = 0 ( a 0 ; q ) n ( a 1 ; q ) n ( a r ; q ) n ( q ; q ) n ( b 1 ; q ) n ( b s ; q ) n ( ( - 1 ) n q ( n 2 ) ) s - r z n .
17.4.5 Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) = m , n 0 ( a ; q ) m + n ( b ; q ) m ( b ; q ) n x m y n ( q ; q ) m ( q ; q ) n ( c ; q ) m + n ,
17.4.6 Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y ) = m , n 0 ( a ; q ) m + n ( b ; q ) m ( b ; q ) n x m y n ( q , c ; q ) m ( q , c ; q ) n ,
17.4.7 Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y ) = m , n 0 ( a , b ; q ) m ( a , b ; q ) n x m y n ( q ; q ) m ( q ; q ) n ( c ; q ) m + n ,
17.4.8 Φ ( 4 ) ( a , b ; c , c ; q ; x , y ) = m , n 0 ( a , b ; q ) m + n x m y n ( q , c ; q ) m ( q , c ; q ) n .
18: 17.7 Special Cases of Higher ϕ s r Functions
17.7.3 ϕ 2 2 ( c 2 / b 2 , b 2 c , c q ; q 2 , q ) = 1 2 ( b 2 , q ; q 2 ) ( c , c q ; q 2 ) ( ( c / b ; q ) ( b ; q ) + ( - c / b ; q ) ( - b ; q ) ) .
17.7.18 ϕ r + 1 r + 2 ( a , b , b 1 q m 1 , , b r q m r b q , b 1 , , b r ; q , a - 1 q 1 - ( m 1 + + m r ) ) = ( q , b q / a ; q ) ( b 1 / b ; q ) m 1 ( b r / b ; q ) m r ( b q , q / a ; q ) ( b 1 ; q ) m 1 ( b r ; q ) m r b m 1 + + m r ,
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 .
17.7.23 ( 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 .
19: 10.31 Power Series
10.31.3 I ν ( z ) I μ ( z ) = ( 1 2 z ) ν + μ k = 0 ( ν + μ + k + 1 ) k ( 1 4 z 2 ) k k ! Γ ( ν + k + 1 ) Γ ( μ + k + 1 ) .
20: 18.1 Notation
q -Pochhammer Symbol
18.1.2 G n ( p , q , x ) = n ! ( n + p ) n P n ( p - q , q - 1 ) ( 2 x - 1 ) ,