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 
:
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(1)
|
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(2)
|
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(3)
|
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(4)
|
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(5)
|
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(6)
|
 |  |  |
(7)
|
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(8)
|
 |  |  |
(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)
|
(of which 
,
, 
, and 
are precisely Appell
hypergeometric functions), and 20 confluent series with 
, 
, and 
not both 2:
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
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(18)
|
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(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).
