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1: 17.2 Calculus
For n = 0 , 1 , 2 , , … …
17.2.6 ( a 1 , a 2 , , a r ; q ) = j = 1 r ( a j ; q ) .
2: 5.2 Definitions
§5.2(iii) Pochhammer’s Symbol
5.2.5 ( a ) n = Γ ( a + n ) / Γ ( a ) , a 0 , - 1 , - 2 , .
5.2.6 ( - a ) n = ( - 1 ) n ( a - n + 1 ) n ,
5.2.7 ( - m ) n = { ( - 1 ) n m ! ( m - n ) ! , 0 n m , 0 , n > m ,
3: 15.15 Sums
15.15.1 F ( a , b c ; 1 z ) = ( 1 - z 0 z ) - a s = 0 ( a ) s s ! F ( - s , b c ; 1 z 0 ) ( 1 - z z 0 ) - s .
4: 17.11 Transformations of q -Appell Functions
17.11.1 Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) = ( a , b x , b y ; q ) ( c , x , y ; q ) ϕ 2 3 ( c / a , x , y b x , b y ; q , a ) ,
17.11.2 Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y ) = ( b , a x ; q ) ( c , x ; q ) n , r 0 ( a , b ; q ) n ( c / b , x ; q ) r b r y n ( q , c ; q ) n ( q ; q ) r ( a x ; q ) n + r ,
17.11.3 Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y ) = ( a , b x ; q ) ( c , x ; q ) n , r 0 ( a , b ; q ) n ( x ; q ) r ( c / a ; q ) n + r a r y n ( q , c / a ; q ) n ( q , b x ; q ) r .
17.11.4 m 1 , , m n 0 ( a ; q ) m 1 + m 2 + + m n ( b 1 ; q ) m 1 ( b 2 ; q ) m 2 ( b n ; q ) m n x 1 m 1 x 2 m 2 x n m n ( q ; q ) m 1 ( q ; q ) m 2 ( q ; q ) m n ( c ; q ) m 1 + m 2 + + m n = ( a , b 1 x 1 , b 2 x 2 , , b n x n ; q ) ( c , x 1 , x 2 , , x n ; q ) ϕ n n + 1 ( c / a , x 1 , x 2 , , x n b 1 x 1 , b 2 x 2 , , b n x n ; q , a ) .
5: 17.12 Bailey Pairs
17.12.3 β n = j = 0 n α j ( q ; q ) n - j ( a q ; q ) n + j .
17.12.4 n = 0 q n 2 a n β n = 1 ( a q ; q ) n = 0 q n 2 a n α n .
6: 17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions
7: 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 ) .
17.3.3 sin q ( x ) = 1 2 i ( e q ( i x ) - e q ( - i x ) ) = n = 0 ( 1 - q ) 2 n + 1 ( - 1 ) n x 2 n + 1 ( q ; q ) 2 n + 1 ,
17.3.4 Sin q ( x ) = 1 2 i ( E q ( i x ) - E q ( - i x ) ) = n = 0 ( 1 - q ) 2 n + 1 q n ( 2 n + 1 ) ( - 1 ) n x 2 n + 1 ( q ; q ) 2 n + 1 .
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 ,
8: 17.8 Special Cases of ψ r r Functions
17.8.1 n = - ( - z ) n q n ( n - 1 ) / 2 = ( q , z , q / z ; q ) ;
17.8.3 n = - ( - 1 ) n q n ( 3 n - 1 ) / 2 z 3 n ( 1 + z q n ) = ( q , - z , - q / z ; q ) ( q z 2 , q / z 2 ; q 2 ) .
17.8.4 ψ 2 2 ( b , c ; a q / b , a q / c ; q , - a q / ( b c ) ) = ( a q / ( b c ) ; q ) ( a q 2 / b 2 , a q 2 / c 2 , q 2 , a q , q / a ; q 2 ) ( a q / b , a q / c , q / b , q / c , - a q / ( b c ) ; q ) ,
17.8.8 ψ 2 2 ( b 2 , b 2 / c q , c q ; q 2 , c q 2 / b 2 ) = 1 2 ( q 2 , q b 2 , q / b 2 , c q / b 2 ; q 2 ) ( c q , c q 2 / b 2 , q 2 / b 2 , c / b 2 ; q 2 ) ( ( c q / b ; q ) ( b q ; q ) + ( - c q / b ; q ) ( - b q ; q ) ) , | c q 2 | < | b 2 | .
9: 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 ) .
10: 18.28 Askey–Wilson Class
18.28.3 2 π sin θ w ( cos θ ) = | ( e 2 i θ ; q ) ( a e i θ , b e i θ , c e i θ , d e i θ ; q ) | 2 ,
18.28.4 h 0 = ( a b c d ; q ) ( q , a b , a c , a d , b c , b d , c d ; 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 , .
18.28.8 1 2 π 0 π Q n ( cos θ ; a , b | q ) Q m ( cos θ ; a , b | q ) | ( e 2 i θ ; q ) ( a e i θ , b e i θ ; q ) | 2 d θ = δ n , m ( q n + 1 , a b q n ; q ) , a , b or a = b ¯ ; | a b | < 1 ; | a | , | b | 1 .
18.28.15 1 2 π 0 π C n ( cos θ ; β | q ) C m ( cos θ ; β | q ) | ( e 2 i θ ; q ) ( β e 2 i θ ; q ) | 2 d θ = ( β , β q ; q ) ( β 2 , q ; q ) ( 1 - β ) ( β 2 ; q ) n ( 1 - β q n ) ( q ; q ) n δ n , m , - 1 < β < 1 .