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1: 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 ) 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 ) .
2: 27.3 Multiplicative Properties
§27.3 Multiplicative Properties
If f is multiplicative, then the values f ( n ) for n > 1 are determined by the values at the prime powers. …Related multiplicative properties are … A function f is completely multiplicative if f ( 1 ) = 1 and … If f is completely multiplicative, then (27.3.2) becomes …
3: 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 ) .
17.7.6 ϕ 2 3 ( q - 2 n , b , c q 1 - 2 n / b , q 1 - 2 n / c ; q , q 2 - n b c ) = ( b , c ; q ) n ( q , b c ; q ) 2 n ( q , b c ; q ) n ( b , c ; q ) 2 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 ,
17.7.22 k = - m n ( 1 - a d p k q k ) ( 1 - b p k / ( d q k ) ) ( 1 - a d ) ( 1 - ( b / d ) ) ( a , b ; p ) k ( c , a d 2 / ( b c ) ; q ) k ( d q , a d q / b ; q ) k ( a d p / c , b c p / d ; p ) k q k = ( 1 - a ) ( 1 - b ) ( 1 - c ) ( 1 - ( a d 2 / ( b c ) ) ) d ( 1 - a d ) ( 1 - ( b / d ) ) ( 1 - ( c / d ) ) ( 1 - ( a d / ( b c ) ) ) ( ( a p , b p ; p ) n ( c q , a d 2 q / ( b c ) ; q ) n ( d q , a d q / b ; q ) n ( a d p / c , b c p / d ; p ) n - ( c / ( a d ) , d / ( b c ) ; p ) m + 1 ( 1 / d , b / ( a d ) ; q ) m + 1 ( 1 / c , b c / ( a d 2 ) ; q ) m + 1 ( 1 / a , 1 / b ; p ) m + 1 ) ,
4: 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.
5: 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 ) ,
17.8.5 ψ 3 3 ( b , c , d q / b , q / c , q / d ; q , q b c d ) = ( q , q / ( b c ) , q / ( b d ) , q / ( c d ) ; q ) ( q / b , q / c , q / d , q / ( b c d ) ; q ) ,
6: 17.10 Transformations of ψ r r Functions
17.10.1 ψ 2 2 ( a , b c , d ; q , z ) = ( a z , d / a , c / b , d q / ( a b z ) ; q ) ( z , d , q / b , c d / ( a b z ) ; q ) ψ 2 2 ( a , a b z / d a z , c ; q , d a ) ,
17.10.2 ψ 2 2 ( a , b c , d ; q , z ) = ( a z , b z , c q / ( a b z ) , d q / ( a b z ) ; q ) ( q / a , q / b , c , d ; q ) ψ 2 2 ( a b z / c , a b z / d a z , b z ; q , c d a b z ) .
17.10.3 ψ 8 8 ( q a 1 2 , - q a 1 2 , c , d , e , f , a q - n , q - n a 1 2 , - a 1 2 , a q / c , a q / d , a q / e , a q / f , q n + 1 , a q n + 1 ; q , a 2 q 2 n + 2 c d e f ) = ( a q , q / a , a q / ( c d ) , a q / ( e f ) ; q ) n ( q / c , q / d , a q / e , a q / f ; q ) n ψ 4 4 ( e , f , a q n + 1 / ( c d ) , q - n a q / c , a q / d , q n + 1 , e f / ( a q n ) ; q , q ) ,
17.10.4 ψ 2 2 ( e , f a q / c , a q / d ; q , a q e f ) = ( q / c , q / d , a q / e , a q / f ; q ) ( a q , q / a , a q / ( c d ) , a q / ( e f ) ; q ) n = - ( 1 - a q 2 n ) ( c , d , e , f ; q ) n ( 1 - a ) ( a q / c , a q / d , a q / e , a q / f ; q ) n ( q a 3 c d e f ) n q n 2 .
17.10.5 ( a q / b , a q / c , a q / d , a q / e , q / ( a b ) , q / ( a c ) , q / ( a d ) , q / ( a e ) ; q ) ( f a , g a , f / a , g / a , q a 2 , q / a 2 ; q ) ψ 8 8 ( q a , - q a , b a , c a , d a , e a , f a , g a a , - a , a q / b , a q / c , a q / d , a q / e , a q / f , a q / g ; q , q 2 b c d e f g ) = ( q , q / ( b f ) , q / ( c f ) , q / ( d f ) , q / ( e f ) , q f / b , q f / c , q f / d , q f / e ; q ) ( f a , q / ( f a ) , a q / f , f / a , g / f , f g , q f 2 ; q ) ϕ 7 8 ( f 2 , q f , - q f , f b , f c , f d , f e , f g f , - f , f q / b , f q / c , f q / d , f q / e , f q / g ; q , q 2 b c d e f g ) + idem ( f ; g ) .
7: 27.20 Methods of Computation: Other Number-Theoretic Functions
To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). …
8: 18.36 Miscellaneous Polynomials
These are OP’s on the interval ( - 1 , 1 ) with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at - 1 and 1 to the weight function for the Jacobi polynomials. …
§18.36(iii) Multiple OP’s
9: 17.9 Further Transformations of ϕ r r + 1 Functions
17.9.6 ϕ 2 3 ( a , b , c d , e ; q , d e / ( a b c ) ) = ( e / a , d e / ( b c ) ; q ) ( e , d e / ( a b c ) ; q ) ϕ 2 3 ( a , d / b , d / c d , d e / ( b c ) ; q , e / a ) ,
17.9.7 ϕ 2 3 ( a , b , c d , e ; q , d e / ( a b c ) ) = ( b , d e / ( a b ) , d e / ( b c ) ; q ) ( d , e , d e / ( a b c ) ; q ) ϕ 2 3 ( d / b , e / b , d e / ( a b c ) d e / ( a b ) , d e / ( b c ) ; q , b ) ,
17.9.11 ϕ 2 3 ( q - n , b , c d , e ; q , q ) = ( e / c , d / c ; q ) n ( e , d ; q ) n c n ϕ 2 3 ( q - n , c , c b q 1 - n / ( d e ) c q 1 - n / e , c q 1 - n / d ; q , q ) ,
17.9.13 ϕ 2 3 ( a , b , c d , e ; q , d e a b c ) = ( e / b , e / c ; q ) ( e , e / ( b c ) ; q ) ϕ 2 3 ( d / a , b , c d , b c q / e ; q , q ) + ( d / a , b , c , d e / ( b c ) ; q ) ( d , e , b c / e , d e / ( a b c ) ; q ) ϕ 2 3 ( e / b , e / c , d e / ( a b c ) d e / ( b c ) , e q / ( b c ) ; q , q ) .
17.9.14 ϕ 3 4 ( q - n , a , b , c d , e , f ; q , q ) = ( e / a , f / a ; q ) n ( e , f ; q ) n a n ϕ 3 4 ( q - n , a , d / b , d / c d , a q 1 - n / e , a q 1 - n / f ; q , q ) = ( a , e f / ( a b ) , e f / ( a c ) ; q ) n ( e , f , e f / ( a b c ) ; q ) n ϕ 3 4 ( q - n , e / a , f / a , e f / ( a b c ) e f / ( a b ) , e f / ( a c ) , q 1 - n / a ; q , q ) .
10: 17.6 ϕ 1 2 Function
17.6.13 ϕ 1 2 ( a , b ; c ; q , q ) + ( q / c , a , b ; q ) ( c / q , a q / c , b q / c ; q ) ϕ 1 2 ( a q / c , b q / c ; q 2 / c ; q , q ) = ( q / c , a b q / c ; q ) ( a q / c , b q / c ; q ) ,
17.6.16 ϕ 1 2 ( a , b c ; q , z ) = ( b , c / a , a z , q / ( a z ) ; q ) ( c , b / a , z , q / z ; q ) ϕ 1 2 ( a , a q / c a q / b ; q , c q / ( a b z ) ) + ( a , c / b , b z , q / ( b z ) ; q ) ( c , a / b , z , q / z ; q ) ϕ 1 2 ( b , b q / c b q / a ; q , c q / ( a b z ) ) , | z | < 1 , | a b z | < | c q | .
17.6.29 ϕ 1 2 ( a , b c ; q , z ) = ( - 1 2 π i ) ( a , b ; q ) ( q , c ; q ) - i i ( q 1 + ζ , c q ζ ; q ) ( a q ζ , b q ζ ; q ) π ( - z ) ζ sin ( π ζ ) d ζ ,
where | z | < 1 , | ph ( - z ) | < π , and the contour of integration separates the poles of ( q 1 + ζ , c q ζ ; q ) / sin ( π ζ ) from those of 1 / ( a q ζ , b q ζ ; q ) , and the infimum of the distances of the poles from the contour is positive. …