Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T12:51:28.338Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalizations to several variables of Lagrange's expansion, with applications to stochastic processes

Published online by Cambridge University Press:  24 October 2008

I. J. Good
I. J. Good
Affiliation:
Admiralty Research Laboratory, Teddington, Middlesex.
Get access Rights & Permissions [Opens in a new window]

Abstract

A generalization to two independent variables of Lagrange's expansion of an inverse function was given by Stieltjes and proved rigorously by Poincaré. A new method of proof is given here that also provides a new and sometimes more convenient form of the generalization. The results are given for an arbitrary number of independent variables. Applications are pointed out to random branching processes, to queues with various types of customers, and to some enumeration problems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 56 , Issue 4 , October 1960 , pp. 367 - 380
Copyright
Copyright © Cambridge Philosophical Society 1960

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Behnke, H. and Thullen, P.Theorie der Funktionen mehrerer komplexen Veränderlichen (Leipzig, 1933).Google Scholar
(2)Blair, C. M. and Henze, H. R.J. Amer. Chem. Soc. 53 (1931), 3042–6 and 3077–85. (Ref. not seen.)Google Scholar
(3)Bochner, S. and Martin, W. T.Several complex variables (Princeton, 1948).Google Scholar
(4)Cayley, A.On the theory of the analytic forms called trees. Phil. Mag. (4) 13 (1857), 172–6.Google Scholar
See also, his Collected mathematical papers (Cambridge, 1889–1897), vol. 3, 242;Google Scholar
(4)Cayley, A.On the theory of the analytic forms called trees. Phil. Mag. (4) 13 (1857), 172–6.CrossRefGoogle Scholar
See also, his Collected mathematical papers (Cambridge, 1889–1897), vol. 9, 202, 427;Google Scholar
(4)Cayley, A.On the theory of the analytic forms called trees. Phil. Mag. (4) 13 (1857), 172–6.Google Scholar
See also, his Collected mathematical papers (Cambridge, 1889–1897), vol. 11, 365;Google Scholar
(4)Cayley, A.On the theory of the analytic forms called trees. Phil. Mag. (4) 13 (1857), 172–6.CrossRefGoogle Scholar
See also, his Collected mathematical papers (Cambridge, 1889–1897), vol. 13, 26.Google Scholar
(5)Dawson, R. and Good, I. J.Exact Markov probabilities from oriented linear graphs. Ann Math. Statist. 28 (1957), 946–56.Google Scholar
(6)Everett, C. J. and Ulam, S.Multiplicative systems in several variables, II (Los Alamos Scientific Laboratory, pp. 23; AECD-2165, LADC-533; declassified 30 July 1948. Part I seems to be ‘out of print’).Google Scholar
(7)Forsyth, A. R.Lectures introductory to the theory of functions of two complex variables (Cambridge, 1914).Google Scholar
(8)Good, I. J.The number of individuals in a cascade process. Proc. Camb. Phil. Soc. 45 (1949), 360–3.Google Scholar
(9)Good, I. J.Contribution to the discussion on D. G. Kendall's paper (16). J. R. Statist. Soc. B, 13 (1951), 182–3.Google Scholar
(10)Good, I. J.The joint distribution of the sizes of the generations in a cascade process. Proc. Camb. Phil. Soc. 51 (1955), 240–2.Google Scholar
(11)Goodman, L. A.Exact probabilities and asymptotic relationships for some statistics for some mth order Markov chains. Ann. Math. Statist. 29 (1958), 476–90.Google Scholar
(12)Goursat, Édouard. A course in mathematical analysis, I and II, Part I. Translated by Hedrick, E. R. and Dunkel, O. (Boston, 1904 and 1916).Google Scholar
(13)Harris, T. E.First passage and recurrence distributions. Trans. Amer. Math. Soc. 73 (1952), 471–83.Google Scholar
(14)Jacobson, N.Lectures in abstract algebra, I (Princeton, 1951).Google Scholar
(15)Jeffreys, H. and Jeffreys, B. S.Methods of mathematical physics (Cambridge, 1946).Google Scholar
(16)Kendall, D. G.The theory of queues. J. R. Statist. Soc. B, 13 (1951), 151–85 (with discussion).Google Scholar
(17)Lagrange, J. L.Mém. Acad. Berlin, 24; (Euvres, II), p. 25. (Ref. not seen.)Google Scholar
(18)Mirsky, L.An introduction to linear algebra (Oxford, 1955).Google Scholar
(19)Otter, R.The number of trees. Ann. Math., Princeton, 49 (1948), 583–99.Google Scholar
(20)Otter, R.The multiplicative process. Ann. Math. Statist. 20 (1949), 206–24.Google Scholar
(21)Poincaré, H.Sur les résidus des intégrates doubles. Acta Math. 9 (1886), 321–80.Google Scholar
(22)Pólya, G.Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und Chemische Verbindungen. Acta Math. 68 (1937), 145–254.Google Scholar
(23)Price, G. B.Review of a paper by C. Massonnet. Math. Rev. 8 (1947), 499.Google Scholar
(24)Sevasty'astov, B. A.The theory of branching random processes. Usp. Matem. Nauk, (N.S.) 6, no. 6 (46) (1951), 47–99. (Russian.)Google Scholar
(25)Stieltjes, T. J. An unpublished manuscript sent to C. Hermite.Google Scholar
(26)Whittaker, E. T.From Euclid to Eddington (Cambridge, 1949).Google Scholar
(27)Whittaker, E. T. and Watson, G. N.A course of modern analysis, 4th edn. (Cambridge, 1935).Google Scholar
(28)Whittle, P.Some distribution and moment formulae for the Markov chain. J. R. Statist. Soc. B, 17 (1955), 235–42.Google Scholar
(29)Whittle, P.Some combinatorial results for matrix powers. Quart. J. Math. (2) 7 (1956), 316–20.CrossRefGoogle Scholar