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1: Bibliography Y
  • Z. M. Yan (1992) Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.
  • 2: 19.21 Connection Formulas
    19.21.1 R F ( 0 , z + 1 , z ) R D ( 0 , z + 1 , 1 ) + R D ( 0 , z + 1 , z ) R F ( 0 , z + 1 , 1 ) = 3 π / ( 2 z ) , z ( - , 0 ] .
    19.21.2 3 R F ( 0 , y , z ) = z R D ( 0 , y , z ) + y R D ( 0 , z , y ) .
    19.21.7 ( x - y ) R D ( y , z , x ) + ( z - y ) R D ( x , y , z ) = 3 R F ( x , y , z ) - 3 y / ( x z ) ,
    19.21.8 R D ( y , z , x ) + R D ( z , x , y ) + R D ( x , y , z ) = 3 ( x y z ) - 1 / 2 ,
    19.21.9 x R D ( y , z , x ) + y R D ( z , x , y ) + z R D ( x , y , z ) = 3 R F ( x , y , z ) .
    3: 16.13 Appell Functions
    §16.13 Appell Functions
    The following four functions of two real or complex variables x and y cannot be expressed as a product of two F 1 2 functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): …
    4: 19.27 Asymptotic Approximations and Expansions
    19.27.7 R D ( x , y , z ) = 3 2 z 3 / 2 ( ln ( 8 z a + g ) - 2 ) ( 1 + O ( a z ) ) , a / z 0 .
    19.27.8 R D ( x , y , z ) = 3 x y z - 6 x y R G ( x , y , 0 ) ( 1 + O ( z g ) ) , z / g 0 .
    19.27.9 R D ( x , y , z ) = 3 x z ( y + z ) ( 1 + O ( b x ln x b ) ) , b / x 0 .
    19.27.10 R D ( x , y , z ) = R D ( 0 , y , z ) - 3 x h z ( 1 + O ( x h ) ) , x / h 0 .
    5: 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 ) . …
    6: 19.20 Special Cases
    19.20.19 R D ( x , y , z ) 3 ( x y z ) - 1 / 2 , z / x y 0 .
    19.20.20 R D ( x , y , y ) = 3 2 ( y - x ) ( R C ( x , y ) - x y ) , x y , y 0 ,
    19.20.21 R D ( x , x , z ) = 3 z - x ( R C ( z , x ) - 1 z ) , x z , x z 0 .
    19.20.22 0 1 t 2 d t 1 - t 4 = 1 3 R D ( 0 , 2 , 1 ) = ( Γ ( 3 4 ) ) 2 ( 2 π ) 1 / 2 = 0.59907 01173 67796 10371 .
    19.20.23 R D ( x , y , a ) = R - 3 4 ( 5 4 , 1 2 ; a 2 , x y ) , a = 1 2 x + 1 2 y .
    7: 19.28 Integrals of Elliptic Integrals
    19.28.3 0 1 t σ - 1 ( 1 - t ) R D ( 0 , t , 1 ) d t = 3 4 σ + 2 ( B ( σ , 1 2 ) ) 2 .
    19.28.5 z R D ( x , y , t ) d t = 6 R F ( x , y , z ) ,
    19.28.6 0 1 R D ( x , y , v 2 z + ( 1 - v 2 ) p ) d v = R J ( x , y , z , p ) .
    8: Viewing DLMF Interactive 3D Graphics
    In the DLMF, we provide facilities for the interactive display of special functions of two independent variables. …
    9: 19.16 Definitions
    A fourth integral that is symmetric in only two variables is defined by
    19.16.5 R D ( x , y , z ) = R J ( x , y , z , z ) = 3 2 0 d t s ( t ) ( t + z ) ,
    19.16.15 R D ( x , y , z ) = R - 3 2 ( 1 2 , 1 2 , 3 2 ; x , y , z ) ,
    10: 19.25 Relations to Other Functions