Horn Function
The 34 distinct convergent hypergeometric series of order two enumerated by Horn (1931) and corrected by Borngässer (1933). There are 14 complete series for which p=p^'=q=q^'=2:
F_1(alpha,beta,beta^',gamma,x,y) = [画像:sum_(m,n)((alpha)_(m+n)(beta)_m(beta^')_n)/((gamma)_(m+n)m!n!)x^my^n]
(1)
F_2(alpha,beta,beta^',gamma,gamma^',x,y) = [画像:sum_(m,n)((alpha)_(m+n)(beta)_m(beta^')_n)/((gamma)_m(gamma^')_nm!n!)x^my^n]
(2)
F_3(alpha,alpha^',beta,beta^',gamma,x,y) = [画像:sum_(m,n)((alpha)_m(alpha^')_n(beta)_m(beta^')_n)/((gamma)_(m+n)m!n!)x^my^n]
(3)
F_4(alpha,beta,gamma,gamma^',x,y) = [画像:sum_(m,n)((alpha)_(m+n)(beta)_(m+n))/((gamma)_m(gamma^')_nm!n!)x^my^n]
(4)
G_1(alpha,beta,beta^',x,y) = [画像:sum_(m,n)((alpha)_(m+n)(beta)_(n-m)(beta^')_(m-n))/(m!n!)x^my^n]
(5)
G_2(alpha,alpha^',beta,beta^',x,y) = [画像:sum_(m,n)((alpha)_m(alpha^')_n(beta)_(n-m)(beta^')_(m-n))/(m!n!)x^my^n]
(6)
G_3(alpha,alpha^',x,y) = [画像:sum_(m,n)((alpha)_(2n-m)(alpha^')_(2m-n))/(m!n!)x^my^n]
(7)
H_1(alpha,beta,gamma,delta,x,y) = [画像:sum_(m,n)((alpha)_(m-n)(beta)_(m+n)(gamma)_n)/((delta)_mm!n!)x^my^n]
(8)
H_2(alpha,beta,gamma,delta,epsilon,x,y) = [画像:sum_(m,n)((alpha)_(m-n)(beta)_m(gamma)_n(delta)_n)/((epsilon)_mm!n!)x^my^n]
(9)
H_3(alpha,beta,gamma,x,y) = [画像:sum_(m,n)((alpha)_(2m+n)(beta)_n)/((gamma)_(m+n)m!n!)x^my^n]
(10)
H_4(alpha,beta,gamma,delta,x,y) = [画像:sum_(m,n)((alpha)_(2m+n)(beta)_n)/((gamma)_m(delta)_nm!n!)x^my^n]
(11)
H_5(alpha,beta,gamma,x,y) = [画像:sum_(m,n)((alpha)_(2m+n)(beta)_(n-m))/((gamma)_nm!n!)x^my^n]
(12)
H_6(alpha,beta,gamma,x,y) = [画像:sum_(m,n)((alpha)_(2m-n)(beta)_(n-m)(gamma)_n)/(m!n!)x^my^n]
(13)
H_7(alpha,beta,gamma,delta,x,y) = [画像:sum_(m,n)((alpha)_(2m-n)(beta)_n(gamma)_n)/((delta)_mm!n!)x^my^n]
(14)
(of which F_1, F_2, F_3, and F_4 are precisely Appell hypergeometric functions), and 20 confluent series with p<=p^'=2, q<=q^'=2, and p,q not both 2:
Phi_1(alpha,beta,gamma,x,y) = [画像:sum_(m,n)((alpha)_(m+n)(beta)_m)/((gamma)_(m+n)m!n!)x^my^n]
(15)
Phi_2(beta,beta^',gamma,x,y) = [画像:sum_(m,n)((beta)_m(beta^')_n)/((gamma)_(m+n)m!n!)x^my^n]
(16)
Phi_3(beta,gamma,x,y) = [画像:sum_(m,n)((beta)_m)/((gamma)_(m+n)m!n!)x^my^n]
(17)
Psi_1(alpha,beta,gamma,gamma^',x,y) = [画像:sum_(m,n)((alpha)_(m+n)(beta)_m)/((gamma)_m(gamma^')_nm!n!)x^my^n]
(18)
Psi_2(alpha,gamma,gamma^',x,y) = [画像:sum_(m,n)((alpha)_(m+n))/((gamma)_m(gamma^')_nm!n!)x^my^n]
(19)
Xi_1(alpha,alpha^',beta,gamma,x,y) = [画像:sum_(m,n)((alpha)_m(alpha^')_n(beta)_m)/((gamma)_(m+n)m!n!)x^my^n]
(20)
Xi_2(alpha,beta,gamma,x,y) = [画像:sum_(m,n)((alpha)_m(beta)_m)/((gamma)_(m+n)m!n!)x^my^n]
(21)
Gamma_1(alpha,beta,beta^',x,y) = [画像:sum_(m,n)((alpha)_m(beta)_(n-m)(beta^')_(m-n))/(m!n!)x^my^n]
(22)
Gamma_2(beta,beta^',x,y) = [画像:sum_(m,n)((beta)_(n-m)(beta^')_(m-n))/(m!n!)x^my^n]
(23)
H_1(alpha,beta,delta,x,y) = [画像:sum_(m,n)((alpha)_(m-n)(beta)_(m+n))/((delta)_mm!n!)x^my^n]
(24)
H_2(alpha,beta,gamma,delta,x,y) = [画像:sum_(m,n)((alpha)_(m-n)(beta)_m(gamma)_n)/((delta)_mm!n!)x^my^n]
(25)
H_3(alpha,beta,delta,x,y) = [画像:sum_(m,n)((alpha)_(m-n)(beta)_m)/((delta)_mm!n!)x^my^n]
(26)
H_4(alpha,gamma,delta,x,y) = [画像:sum_(m,n)((alpha)_(m-n)(gamma)_n)/((delta)_mm!n!)x^my^n]
(27)
H_5(alpha,delta,x,y) = [画像:sum_(m,n)((alpha)_(m-n))/((delta)_mm!n!)x^my^n]
(28)
H_6(alpha,gamma,x,y) = [画像:sum_(m,n)((alpha)_(2m+n))/((gamma)_(m+n)m!n!)x^my^n]
(29)
H_7(alpha,gamma,delta,x,y) = [画像:sum_(m,n)((alpha)_(2m+n))/((gamma)_m(delta)_nm!n!)x^my^n]
(30)
H_8(alpha,beta,x,y) = [画像:sum_(m,n)((alpha)_(2m-n)(beta)_(n-m))/(m!n!)x^my^n]
(31)
H_9(alpha,beta,delta,x,y) = [画像:sum_(m,n)((alpha)_(2m-n)(beta)_n)/((delta)_mm!n!)x^my^n]
(32)
H_(10)(alpha,delta,x,y) = [画像:sum_(m,n)((alpha)_(2m-n))/((delta)_mm!n!)x^my^n]
(33)
H_(11)(alpha,beta,gamma,delta,x,y) = [画像:sum_(m,n)((alpha)_(m-n)(beta)_n(gamma)_n)/((delta)_mm!n!)x^my^n]
(34)
(Erdélyi et al. 1981, pp. 224-226; Srivastava and Karlsson 1985, pp. 24-26). Here, the sums are taken over nonnegative integers m and n.
Note that Phi_1, Phi_2, and Xi_2 as defined by Erdélyi et al. (1981) are erroneous; the correct formulas given above may be found in Srivastava and Karlsson (1985, pp. 25-26).
See also
Appell Hypergeometric Function, Kampé de Fériet Function, Lauricella FunctionsExplore with Wolfram|Alpha
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References
Borngässer, L. Über hypergeometrische Funktionen zweier Veränderlichen. Dissertation. Darmstadt, Germany: University of Darmstadt, 1933.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Horn's List" and "Convergence of the Series." §5.7.1 and 5.7.2 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 224-229, 1981.Horn, J. "Hypergeometrische Funktionen zweier Veränderlichen." Math. Ann. 105, 381-407, 1931.Srivastava, H. M. and Karlsson, P. W. Multiple Gaussian Hypergeometric Series. Chichester, England: Ellis Horwood, 1985.Referenced on Wolfram|Alpha
Horn FunctionCite this as:
Weisstein, Eric W. "Horn Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HornFunction.html