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 :

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)

(of which , , , and  are precisely Appell hypergeometric functions), and 20 confluent series with , , and  not both 2:

			(15)
			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)
			(27)
			(28)
			(29)
			(30)
			(31)
			(32)
			(33)
			(34)

(Erdélyi et al. 1981, pp. 224-226; Srivastava and Karlsson 1985, pp. 24-26). Here, the sums are taken over nonnegative integers  and .

Note that , , and  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).

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.