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1: 17.11 Transformations of q -Appell Functions
β–Ί
17.11.1 Ξ¦ ( 1 ) ⁑ ( a ; b , b ; c ; q ; x , y ) = ( a , b ⁒ x , b ⁒ y ; q ) ( c , x , y ; q ) ⁒ Ο• 2 3 ⁑ ( c / a , x , y b ⁒ x , b ⁒ y ; q , a ) ,
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17.11.2 Ξ¦ ( 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 ,
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17.11.3 Ξ¦ ( 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 .
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17.11.4 m 1 , , m n ≧ 0 ( a ; q ) m 1 + m 2 + β‹― + m n ⁒ ( b 1 ; q ) m 1 ⁒ ( b 2 ; q ) m 2 ⁒ β‹― ⁒ ( b n ; q ) m n ⁒ x 1 m 1 ⁒ x 2 m 2 ⁒ β‹― ⁒ x n m n ( q ; q ) m 1 ⁒ ( q ; q ) m 2 ⁒ β‹― ⁒ ( q ; q ) m n ⁒ ( c ; q ) m 1 + m 2 + β‹― + m n = ( a , b 1 ⁒ x 1 , b 2 ⁒ x 2 , , b n ⁒ x n ; q ) ( c , x 1 , x 2 , , x n ; q ) ⁒ Ο• 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
β–ΊIf f is multiplicative, then the values f ⁑ ( n ) for n > 1 are determined by the values at the prime powers. …Related multiplicative properties are … β–ΊA function f is completely multiplicative if f ⁑ ( 1 ) = 1 and … β–ΊIf f is completely multiplicative, then (27.3.2) becomes …
3: 17.7 Special Cases of Higher Ο• s r Functions
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17.7.2 Ο• 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 ) .
β–Ί β–Ί
17.7.6 Ο• 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 .
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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 ,
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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
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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 ) ) ,
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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: 17.10 Transformations of ψ r r Functions
β–Ί
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 ) ,
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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 ) .
β–Ί
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 ) ,
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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 .
β–Ί
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 ) ⁒ Ο• 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 ) .
6: 17.8 Special Cases of ψ r r Functions
β–Ί
17.8.1 n = ( z ) n ⁒ q n ⁒ ( n 1 ) / 2 = ( q , z , q / z ; q ) ;
β–Ί
17.8.2 ψ 1 1 ⁑ ( a b ; q , z ) = ( q , b / a , a ⁒ z , q / ( a ⁒ z ) ; q ) ( b , q / a , z , b / ( a ⁒ z ) ; q ) , | b / a | < | z | < 1 .
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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 ) .
β–Ί
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 ) , | q ⁒ a | < | b ⁒ c | ,
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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 ) , | q | < | b ⁒ c ⁒ d | ,
7: 17.9 Further Transformations of Ο• r r + 1 Functions
β–Ί
17.9.3_5 Ο• 1 2 ⁑ ( a , b c ; q , z ) = ( c / a , c / b ; q ) ( c , c / ( a ⁒ b ) ; q ) ⁒ Ο• 2 3 ⁑ ( a , b , a ⁒ b ⁒ z / c q ⁒ a ⁒ b / c , 0 ; q , q ) + ( a , b , a ⁒ b ⁒ z / c ; q ) ( c , a ⁒ b / c , z ; q ) ⁒ Ο• 2 3 ⁑ ( c / a , c / b , z q ⁒ c / ( a ⁒ b ) , 0 ; q , q ) ,
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17.9.6 Ο• 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 ) ⁒ Ο• 2 3 ⁑ ( a , d / b , d / c d , d ⁒ e / ( b ⁒ c ) ; q , e / a ) ,
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17.9.7 Ο• 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 ) ⁒ Ο• 2 3 ⁑ ( d / b , e / b , d ⁒ e / ( a ⁒ b ⁒ c ) d ⁒ e / ( a ⁒ b ) , d ⁒ e / ( b ⁒ c ) ; q , b ) ,
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17.9.13 Ο• 2 3 ⁑ ( a , b , c d , e ; q , d ⁒ e a ⁒ b ⁒ c ) = ( e / b , e / c ; q ) ( e , e / ( b ⁒ c ) ; q ) ⁒ Ο• 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 ) ⁒ Ο• 2 3 ⁑ ( e / b , e / c , d ⁒ e / ( a ⁒ b ⁒ c ) d ⁒ e / ( b ⁒ c ) , e ⁒ q / ( b ⁒ c ) ; q , q ) .
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17.9.14 Ο• 3 4 ⁑ ( q n , a , b , c d , e , f ; q , q ) = ( e / a , f / a ; q ) n ( e , f ; q ) n ⁒ a n ⁒ Ο• 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 ⁒ Ο• 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 ) .
8: 27.20 Methods of Computation: Other Number-Theoretic Functions
β–ΊTo 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). …
9: 17.6 Ο• 1 2 Function
β–Ί
17.6.1 Ο• 1 2 ⁑ ( a , b c ; q , c / ( a ⁒ b ) ) = ( c / a , c / b ; q ) ( c , c / ( a ⁒ b ) ; q ) , | c | < | a ⁒ b | .
β–Ί
17.6.5 Ο• 1 2 ⁑ ( a , b a ⁒ q / b ; q , q / b ) = ( q ; q ) ⁒ ( a ⁒ q , a ⁒ q 2 / b 2 ; q 2 ) ( q / b , a ⁒ q / b ; q ) , | b | > | q | .
β–Ί
17.6.13 Ο• 1 2 ⁑ ( a , b ; c ; q , q ) + ( q / c , a , b ; q ) ( c / q , a ⁒ q / c , b ⁒ q / c ; q ) ⁒ Ο• 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 ) ,
β–Ί
17.6.29 Ο• 1 2 ⁑ ( a , b c ; q , z ) = ( 1 2 ⁒ Ο€ ⁒ i ) ⁒ ( a , b ; q ) ( q , c ; q ) ⁒ i ⁒ i ⁒ ( q 1 + ΞΆ , c ⁒ q ΞΆ ; q ) ( a ⁒ q ΞΆ , b ⁒ q ΞΆ ; q ) ⁒ Ο€ ⁒ ( z ) ΞΆ sin ⁑ ( Ο€ ⁒ ΞΆ ) ⁒ d ΞΆ ,
β–Ίwhere | z | < 1 , | ph ⁑ ( z ) | < Ο€ , and the contour of integration separates the poles of ( q 1 + ΞΆ , c ⁒ q ΞΆ ; q ) / sin ⁑ ( Ο€ ⁒ ΞΆ ) from those of 1 / ( a ⁒ q ΞΆ , b ⁒ q ΞΆ ; q ) , and the infimum of the distances of the poles from the contour is positive. …
10: 17.4 Basic Hypergeometric Functions
β–Ί
17.4.3 ψ s r ⁑ ( a 1 , a 2 , , a r b 1 , b 2 , , b s ; q , z ) = ψ s r ⁑ ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) = n = ( a 1 , a 2 , , a r ; q ) n ⁒ ( 1 ) ( s r ) ⁒ n ⁒ q ( s r ) ⁒ ( n 2 ) ⁒ z n ( b 1 , b 2 , , b s ; q ) n = n = 0 ( a 1 , a 2 , , a r ; q ) n ⁒ ( 1 ) ( s r ) ⁒ n ⁒ q ( s r ) ⁒ ( n 2 ) ⁒ z n ( b 1 , b 2 , , b s ; q ) n + n = 1 ( q / b 1 , q / b 2 , , q / b s ; q ) n ( q / a 1 , q / a 2 , , q / a r ; q ) n ⁒ ( b 1 ⁒ b 2 ⁒ β‹― ⁒ b s a 1 ⁒ a 2 ⁒ β‹― ⁒ a r ⁒ z ) n .
β–Ί
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 ,
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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 ,
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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 .