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1: 17.7 Special Cases of Higher ϕ s r Functions
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 .
Continued Fractions
For continued-fraction representations of a ratio of ϕ 2 3 functions, see Cuyt et al. (2008, pp. 399–400). …
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 ) ,
2: 12.6 Continued Fraction
§12.6 Continued Fraction
For 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 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 ) ,
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.
§17.6(vi) Continued Fractions
For continued-fraction representations of the ϕ 1 2 function, see Cuyt et al. (2008, pp. 395–399).
4: 27.19 Methods of Computation: Factorization
These 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
17.2.6 ( a 1 , a 2 , , a r ; q ) = j = 1 r ( a j ; q ) .
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 ,
17.2.23 ( q a 1 k , q ω k a 1 k , , q ω k k - 1 a 1 k ; q ) n ( a 1 k , ω k a 1 k , , ω 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
For 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
Stieltjes Fractions
is called a Stieltjes fraction ( S -fraction). …
Jacobi Fractions
The continued fraction
8: 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 ) .
9: 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 …
10: 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 ) ,