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)
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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)
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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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(of which , , , and are precisely Appell hypergeometric functions), and 20 confluent series with , , and not both 2:
(15)
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(16)
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(17)
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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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(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).