The need for a new classification of double hypergeometric series

Carlson, B. C.

doi:10.1090/S0002-9939-1976-0402138-8

The need for a new classification of double hypergeometric series
HTML articles powered by AMS MathViewer

by B. C. Carlson PDF

Proc. Amer. Math. Soc. 56 (1976), 221-224 Request permission

Abstract:

An elementary proof is given of a linear transformation which changes a particular double hypergeometric series of order two into a series of order three. A similar transformation holds for a particular series of arbitrary order in two or more variables. The change in order provides new evidence, the most compelling to date, that the order of such a series is not a fundamental property. This conclusion undermines the accepted classification of hypergeometric series in more than one variable.

References

B. C. Carlson, Appell’s function $F_{4}$ as a double average, SIAM J. Math. Anal. 6 (1975), no. 6, 960–965. MR 385184, DOI 10.1137/0506084
B. C. Carlson, Quadratic transformations of Appell functions, SIAM J. Math. Anal. 7 (1976), no. 2, 291–304. MR 393593, DOI 10.1137/0507023
A. Erdélyi, Transformations of hypergeometric functions of two variables, Proc. Roy. Soc. Edinburgh Sect. A 62 (1948), 378–385. MR 26160
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
J. Horn, Ueber die Convergenz der hypergeometrischen Reihen zweier und dreier Veränderlichen, Math. Ann. 34 (1889), no. 4, 544–600 (German). MR 1510591, DOI 10.1007/BF01443681
J. Horn, Hypergeometrische Funktionen zweier Veränderlichen, Math. Ann. 105 (1931), no. 1, 381–407 (German). MR 1512721, DOI 10.1007/BF01455825