About the Project

pochhammer

AdvancedHelp

(0.001 seconds)

1—10 of 132 matching pages

1: 17.14 Constant Term Identities
17.14.1 ( q ; q ) a 1 + a 2 + + a n ( q ; q ) a 1 ( q ; q ) a 2 ( q ; q ) a n =  coeff. of  x 1 0 x 2 0 x n 0  in  1 j < k n ( x j x k ; q ) a j ( q x k x j ; q ) a k .
17.14.2 n = 0 q n ( n + 1 ) ( q 2 ; q 2 ) n ( q ; q 2 ) n + 1 =  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 1 q 2 ; q 2 ) ( q ; q 2 ) ( z 1 q ; q 2 ) = 1 ( q ; q 2 )  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 1 q ; q ) = H ( q ) ( q ; q 2 ) ,
17.14.3 n = 0 q n ( n + 1 ) ( q 2 ; q 2 ) n ( q ; q 2 ) n + 1 =  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 1 ; q 2 ) ( q ; q 2 ) ( z 1 q ; q 2 ) = 1 ( q ; q 2 )  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 1 ; q ) = G ( q ) ( q ; q 2 ) ,
17.14.4 n = 0 q n 2 ( q 2 ; q 2 ) n ( q ; q 2 ) n =  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 1 ; q 2 ) ( q ; q 2 ) ( z 1 ; q 2 ) = 1 ( q ; q 2 )  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( z 2 ; q 4 ) = G ( q 4 ) ( q ; q 2 ) ,
17.14.5 n = 0 q n 2 + 2 n ( q 2 ; q 2 ) n ( q ; q 2 ) n + 1 =  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( q 2 z 1 ; q 2 ) ( q ; q 2 ) ( z 1 q 2 ; q 2 ) = 1 ( q ; q 2 )  coeff. of  z 0  in  ( z q ; q 2 ) ( z 1 q ; q 2 ) ( q 2 ; q 2 ) ( q 4 z 2 ; q 4 ) = H ( q 4 ) ( q ; q 2 ) .
2: 5.2 Definitions
§5.2(iii) Pochhammer’s Symbol
( a ) 0 = 1 ,
( a ) 2 n = 2 2 n ( a 2 ) n ( a + 1 2 ) n ,
( a ) 2 n + 1 = 2 2 n + 1 ( a 2 ) n + 1 ( a + 1 2 ) n .
Pochhammer symbols (rising factorials) ( x ) n = x ( x + 1 ) ( x + n 1 ) and falling factorials ( 1 ) n ( x ) n = x ( x 1 ) ( x n + 1 ) can be expressed in terms of each other via …
3: 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 ) .
4: 16.13 Appell Functions
16.13.1 F 1 ( α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m ( β ) n ( γ ) m + n m ! n ! x m y n , max ( | x | , | y | ) < 1 ,
16.13.2 F 2 ( α ; β , β ; γ , γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m ( β ) n ( γ ) m ( γ ) n m ! n ! x m y n , | x | + | y | < 1 ,
16.13.3 F 3 ( α , α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m ( α ) n ( β ) m ( β ) n ( γ ) m + n m ! n ! x m y n , max ( | x | , | y | ) < 1 ,
16.13.4 F 4 ( α , β ; γ , γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m + n ( γ ) m ( γ ) n m ! n ! x m y n , | x | + | y | < 1 .
5: 17.2 Calculus
For n = 0 , 1 , 2 , , … … For properties of the function f ( q ) = q 1 / 24 η ( ln q 2 π i ) = ( q ; q ) see §27.14. …
17.2.18 ( a q k ; q ) n k = ( a ; q ) n ( a ; q ) k .
17.2.21 ( a 2 ; q 2 ) n = ( a ; q ) n ( a ; q ) n ,
6: 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 ) .
7: 16.1 Special Notation
p , q nonnegative integers.
( 𝐚 ) k ( a 1 ) k ( a 2 ) k ( a p ) k .
( 𝐛 ) k ( b 1 ) k ( b 2 ) k ( b q ) k .
8: 16.9 Zeros
Next, assume that p = q and that the a j and the quotients ( 𝐚 ) j / ( 𝐛 ) j are all real. …
9: 17.9 Further Transformations of ϕ r r + 1 Functions
17.9.3 ϕ 1 2 ( a , b c ; q , z ) = ( a b z / c ; q ) ( b z / c ; q ) ϕ 2 3 ( a , c / b , 0 c , c q / ( b z ) ; q , q ) + ( a , b z , c / b ; q ) ( c , z , c / ( b z ) ; q ) ϕ 2 3 ( z , a b z / c , 0 b z , b z q / c ; q , q ) ,
17.9.4 ϕ 1 2 ( q n , b c ; q , z ) = ( c / b ; q ) n ( c ; q ) n ( b z q ) n ϕ 2 3 ( q n , q / z , q 1 n / c b q 1 n / c , 0 ; q , q ) ,
17.9.19 n = 0 ( a ; q 2 ) n ( b ; q ) n ( q 2 ; q 2 ) n ( c ; q ) n z n = ( b ; q ) ( a z ; q 2 ) ( c ; q ) ( z ; q 2 ) n = 0 ( c / b ; q ) 2 n ( z ; q 2 ) n b 2 n ( q ; q ) 2 n ( a z ; q 2 ) n + ( b ; q ) ( a z q ; q 2 ) ( c ; q ) ( z q ; q 2 ) n = 0 ( c / b ; q ) 2 n + 1 ( z q ; q 2 ) n b 2 n + 1 ( q ; q ) 2 n + 1 ( a z q ; q 2 ) n .
17.9.20 n = 0 ( a ; q k ) n ( b ; q ) k n z n ( q k ; q k ) n ( c ; q ) k n = ( b ; q ) ( a z ; q k ) ( c ; q ) ( z ; q k ) n = 0 ( c / b ; q ) n ( z ; q k ) n b n ( q ; q ) n ( a z ; q k ) n , k = 1 , 2 , 3 , .
10: 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 .
( a q ρ 1 , a q ρ 2 ; q ) n β n = j = 0 n ( ρ 1 , ρ 2 ; q ) j ( a q ρ 1 ρ 2 ; q ) n j ( a q ρ 1 ρ 2 ) j β j ( q ; q ) n j
α n = ( a ; q ) n ( 1 a q 2 n ) ( 1 ) n q n ( 3 n 1 ) / 2 a n ( q ; q ) n ( 1 a ) ,
β n = 1 ( q ; q ) n .