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Appell functions

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1: 16.13 Appell Functions
§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 ,
16.13.4 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 .
2: 16.24 Physical Applications
§16.24 Physical Applications
§16.24(i) Random Walks
Generalized hypergeometric functions and Appell functions appear in the evaluation of the so-called Watson integrals which characterize the simplest possible lattice walks. …
§16.24(ii) Loop Integrals in Feynman Diagrams
Appell functions are used for the evaluation of one-loop integrals in Feynman diagrams. …
3: 16.16 Transformations of Variables
§16.16 Transformations of Variables
§16.16(i) Reduction Formulas
16.16.3 F 2 ( α ; β , β ; γ , α ; x , y ) = ( 1 - y ) - β F 1 ( β ; α - β , β ; γ ; x , x 1 - y ) ,
For quadratic transformations of Appell functions see Carlson (1976).
4: 16.15 Integral Representations and Integrals
§16.15 Integral Representations and Integrals
16.15.1 F 1 ( α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( α ) Γ ( γ - α ) 0 1 u α - 1 ( 1 - u ) γ - α - 1 ( 1 - u x ) β ( 1 - u y ) β d u , α > 0 , ( γ - α ) > 0 ,
These representations can be used to derive analytic continuations of the Appell functions, including convergent series expansions for large x , large y , or both. For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).
5: 17.1 Special Notation
The main functions treated in this chapter are the basic hypergeometric (or q -hypergeometric) function ϕ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , the bilateral basic hypergeometric (or bilateral q -hypergeometric) function ψ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , and the q -analogs of the Appell functions Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) , Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y ) , Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y ) , and Φ ( 4 ) ( a , b ; c , c ; q ; x , y ) . …
6: 17.11 Transformations of q -Appell Functions
§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 .
7: 16.14 Partial Differential Equations
§16.14(i) Appell Functions
In addition to the four Appell functions there are 24 other sums of double series that cannot be expressed as a product of two F 1 2 functions, and which satisfy pairs of linear partial differential equations of the second order. …
8: 17.4 Basic Hypergeometric Functions
§17.4(iii) q -Appell Functions
17.4.5 Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) = m , n 0 ( a ; q ) m + n ( b ; q ) m ( b ; q ) n x m y n ( q ; q ) m ( q ; q ) n ( c ; q ) m + n ,
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 ; q ) m ( q ; q ) n ( c ; q ) m ( c ; q ) n ,
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 ,
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 .
9: 19.5 Maclaurin and Related Expansions
§19.5 Maclaurin and Related Expansions
where F 1 ( α ; β , β ; γ ; x , y ) is an Appell function16.13). …
10: 16.1 Special Notation
The main functions treated in this chapter are the generalized hypergeometric function F q p ( a 1 , , a p b 1 , , b q ; z ) , the Appell (two-variable hypergeometric) functions F 1 ( α ; β , β ; γ ; x , y ) , F 2 ( α ; β , β ; γ , γ ; x , y ) , F 3 ( α , α ; β , β ; γ ; x , y ) , F 4 ( α , β ; γ , γ ; x , y ) , and the Meijer G -function G p , q m , n ( z ; a 1 , , a p b 1 , , b q ) . Alternative notations are F q p ( a b ; z ) , F q p ( a 1 , , a p ; b 1 , , b q ; z ) , and F q p ( a ; b ; z ) for the generalized hypergeometric function, F 1 ( α , β , β ; γ ; x , y ) , F 2 ( α , β , β ; γ , γ ; x , y ) , F 3 ( α , α , β , β ; γ ; x , y ) , F 4 ( α , β ; γ , γ ; x , y ) , for the Appell functions, and G p , q m , n ( z ; a ; b ) for the Meijer G -function.