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In the theory of special functions in mathematics, the Horn functions (named for Jakob Horn) are the 34 distinct convergent hypergeometric series of order two (i.e. having two independent variables), enumerated by Horn (1931) (corrected by Borngässer (1933)). They are listed in (Erdélyi 1953, section 5.7.1). B. C. Carlson[1] revealed a problem with the Horn function classification scheme.[2]

References

'Profile: Bille C. Carlson' in Digital Library of Mathematical Functions. National Institute of Standards and Technology.

Carlson, B. C. (1976). "The need for a new classification of double hypergeometric series". Proc. Amer. Math. Soc. 56: 221–224. MR 0402138. doi:10.1090/s0002-9939-1976-0402138-8.

Borngässer, Ludwig (1933), Über hypergeometrische funkionen zweier Veränderlichen, Dissertation, Darmstadt
Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol I (PDF), McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0058756
Horn, J. (1935), "Hypergeometrische Funktionen zweier Veränderlichen", Mathematische Annalen, 105 (1): 381–407, doi:10.1007/BF01455825
J. Horn Math. Ann. 111, 637 (1933)
Srivastava, H. M.; Karlsson, Per W. (1985), Multiple Gaussian hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-85312-602-7, MR 834385

Functions