Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The differential equations of birth-and-death processes, and the Stieltjes moment problem


Authors: S. Karlin and J. L. McGregor
Journal: Trans. Amer. Math. Soc. 85 (1957), 489-546
MSC: Primary 60.0X
DOI: https://doi.org/10.1090/S0002-9947-1957-0091566-1
MathSciNet review: 0091566
Full-text PDF Free Access

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [1] J. Favard, Sur les polynomes de Tchebicheff, C. R. (Doklady) Acad. Sci. URSS. vol. 200 (1935) pp. 2052-2053.
  • [2] W. Feller, The parabolic differential equations and the associated semigroups of transformations, Ann. of Math. vol. 55 (1952) pp. 468-519. MR 0047886 (13:948a)
  • [3] -, Diffusion processes in one dimension, Trans. Amer. Math. Soc. vol. 77 (1954) pp. 1-31. MR 0063607 (16:150d)
  • [4] -, An introduction to probability theory and its applications, John Wiley and Sons, 1950.
  • [5] V. Gantmacher, and M. Krein, Sur les matrices complétement non négatives et oscillatoires, Compositio Math. vol. 4 (1937) pp. 445-476. MR 1556987
  • [6] E. Hille, The abstract Cauchy problem and Cauchy's problem for parabolic differential equations, Journal d'Analyse Mathématique vol. 3 (1953-1954) pp. 83-196. MR 0062934 (16:45a)
  • [7] M. Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly vol. 54 (1947) pp. 369-391. MR 0021262 (9:46c)
  • [8] S. Karlin and J. L. McGregor, Representation of a class of stochastic processes, Proc. Nat. Acad. Sci. U.S.A. vol. 41 (1955) pp. 387-391. MR 0071665 (17:166e)
  • [9] W. Lederman, and G. E. H. Reuter, Spectral theory for the differential equations of simple birth and death processes, Philos. Trans. Roy. Soc. London vol. 246 (1954) pp. 321-369. MR 0060103 (15:625g)
  • [10] C. Loewner, oral communication.
  • [11] G. E. H. Reuter and W. Lederman, On the differential equations for the transition probabilities of Markov processes with enumerably many states, Proc. Cambridge Philos. Soc. vol. 49 (1953) pp. 247-262. MR 0053343 (14:757e)
  • [12] J. A. Shohat and J. D. Tamarkin, The problem of moments, Mathematical Surveys, vol. 1, 1943. MR 0008438 (5:5c)
  • [13] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloquium Publications, vol. 23, 1939.
  • [14] D. Widder, The Laplace transform, Princeton University Press, 1941. MR 0005923 (3:232d)
  • [15] I. S. Schoenberg, On Pólya frequency functions, Journal d'Analyse Mathématique vol. 1 (1951).
  • [16] I. I. Hirschman, Jr. and D. Widder, The convolution transform, Princeton University Press, 1955. MR 0073746 (17:479c)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60.0X

Retrieve articles in all journals with MSC: 60.0X


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1957-0091566-1
Article copyright: © Copyright 1957 American Mathematical Society

American Mathematical Society