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1: 17.7 Special Cases of Higher Ο• s r Functions
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 .
β–Ί
Continued Fractions
β–ΊFor continued-fraction representations of a ratio of Ο• 2 3 functions, see Cuyt et al. (2008, pp. 399–400). … β–Ί
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 ,
β–Ί
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 ) ,
2: 12.6 Continued Fraction
§12.6 Continued Fraction
β–ΊFor a continued-fraction expansion of the ratio U ⁑ ( a , x ) / U ⁑ ( a 1 , x ) see Cuyt et al. (2008, pp. 340–341).
3: 17.6 Ο• 1 2 Function
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. β–Ί
§17.6(vi) Continued Fractions
β–ΊFor continued-fraction representations of the Ο• 1 2 function, see Cuyt et al. (2008, pp. 395–399).
4: 27.19 Methods of Computation: Factorization
β–ΊThese algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs). …
5: 17.2 Calculus
β–Ίβ–Ί β–Ί
17.2.6 ( a 1 , a 2 , , a r ; q ) = j = 1 r ( a j ; q ) .
β–Ί
17.2.22 ( q ⁒ a 1 2 , q ⁒ a 1 2 ; q ) n ( a 1 2 , a 1 2 ; q ) n = ( a ⁒ q 2 ; q 2 ) n ( a ; q 2 ) n = 1 a ⁒ q 2 ⁒ n 1 a ,
β–Ί
17.2.23 ( q ⁒ a 1 k , q ⁒ Ο‰ k ⁒ a 1 k , , q ⁒ Ο‰ k k 1 ⁒ a 1 k ; q ) n ( a 1 k , Ο‰ k ⁒ a 1 k , , Ο‰ k k 1 ⁒ a 1 k ; q ) n = ( a ⁒ q k ; q k ) n ( a ; q k ) n = 1 a ⁒ q k ⁒ n 1 a ,
6: 10.55 Continued Fractions
§10.55 Continued Fractions
β–ΊFor continued fractions for 𝗃 n + 1 ⁑ ( z ) / 𝗃 n ⁑ ( z ) and 𝗂 n + 1 ( 1 ) ⁑ ( z ) / 𝗂 n ( 1 ) ⁑ ( z ) see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).
7: 3.10 Continued Fractions
§3.10 Continued Fractions
β–Ί
Stieltjes Fractions
β–Ίis called a Stieltjes fraction ( S -fraction). … β–Ί
Jacobi Fractions
β–ΊThe continued fraction
8: 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 ) ,
β–Ί
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 ,
β–Ί
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 .
β–Ί
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 ) .
9: 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 …
10: 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 β–Ί
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 ) ,
β–Ί
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 ) ,