On the generating functions of totally positive sequences I

The related concept of ann-times positive sequence was first introduced by M. Fekete; see M. Fekete and G. Pólya, “Ueber ein Problem von Laguerre”,Rendiconti di Palermo, 34 (1912), 1–32, especially pages 3–7. These remarks may be found in I. J. Schoenberg, Some analytical aspects of the problem of smoothing,Courant Anniversary Volume, New York 1948, pp. 351–370, especially §2. The problem solved by Theorem 1 is of the type discussed by J. Hadamard in his classical dissertation, “Essai sur l'étude des fonctions données par leur développement de Taylor”,Journal de Mathématiques, 4th series, 8 (1892). Albert Edrei, On the generating functions of totally positive sequences, II, this Journal, p. 104. See also the note by Michael Aissen, Albert Edrei, I. J. Schoenberg and Anne Whitney, On the generating functions of totally positive sequences,Proceedings of the National Academy of Sciences, 37 (1951), pp. 303–307, where the main results of these notes are stated and several applications are discussed. See M. Bôcher, Introduction to Higher Algebra, Chapter II, §11. A. M. Whitney, A reduction theorem for totally positive matrices, this Journal pp. 88–92.