A Horn function is a multiple power series
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(1)
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for which, for each 
,
..., 
,
the ratio of neighboring coefficients is a rational
function of the summation indices,
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(2)
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where 
and 
are polynomials.
The series is called complete if, for each 
, ..., 
, the limits
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(3)
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exist and are neither identically zero nor equal to infinity (Horn 1931).
Horn (1931) enumerated 34 distinct two-variable hypergeometric series that converge in nonempty regions, with
corrections by Borngässer (1933). There are 14 complete series for which 
:
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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(13)
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(14)
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(15)
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(16)
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(17)
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(of which 
,
, 
, and 
are precisely Appell
hypergeometric functions), and 20 confluent series with 
, 
, and 
not both 2:
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(18)
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(19)
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(20)
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(21)
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(22)
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(23)
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(24)
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(25)
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(26)
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(27)
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(28)
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(29)
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(30)
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(31)
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(32)
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(33)
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(34)
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(35)
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(36)
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(37)
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(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).
The rational coefficient ratios determine an associated Horn system that annihilates the series.
