Asymptotic Representations of Confluent Hypergeometric Functions.

I A more general theory will result if in the place of R we employ an abstract normed ring. s We use the symbols =, ... ... in more than one sense. No confusion need arise as tie context makes clear the meaning of each such symbol. It is worth while to mention here that the relation of equality = for E1 as well as for E2 is not an independent primitive idea; for, an equivalent set of postulates can be given in which the equality = for E1 as well as for E2 is defined by means of the properties of the corresponding norm function 6 Banach, S., Theorie des Operations Line'aires, chapter IV, Warsaw, 1932; see also Frechet, M., Espaces Abstraits, Paris, 1928. 7 Michal, A. D., "Affinely Connected Function Space Manifolds," Am. Jour. Math., 50, 473-517, especially 509-517 (1928); "Differential Geometries of Function Space," these PROCEEDINGS, 16, 88-94 (Jan., 1930). See also Peterson, T. S., "The Analogue of Weyl's Conformal Curvature Tensor in a Michal Functional Geometry," Annali di Mat., 13, 55-62 (1934). 8 The first partial differential with respect to x of a function F(x, Yi. yn) is denoted by F(x, yi, ... yn; 5x). All differentials are taken in the Frechet sense.