Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-08T03:24:18.924Z Has data issue: false hasContentIssue false

Random motion and analytic continued fractions

Published online by Cambridge University Press:  24 October 2008

I. J. Good
Affiliation:
25 Scott House Princess Elizabeth Way Cheltenham

Extract

In a previous note (Good (7)) it was shown that there is an intimate connexion between Legendre polynomials and ‘trinomial’ random walks, that is walks on a one-dimensional lattice with the same probabilities p–1p0, p1 at each point of taking a step of – 1,0,1. The purpose of the present note is to point out that if the probabilities depend on the current position of the particle, then there is scope for applications of the theory of analytic continued fractions.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1958

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)Bateman Project Staff. Higher transcendental functions, III (New York, 1955).Google Scholar
(2)Euler, L.Introductio in analysin infinitorum, I (1748), chapter 18.Google Scholar
(3)Feller, W.Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67 (1949), 98119.CrossRefGoogle Scholar
(4)Feller, W.An introduction to probability theory and ita opplications, I (New York, 1950).Google Scholar
(5)Gauss, C.F. Disquisitiones generales circa seriem infinitam. Werke, III (1876), 134–8.Google Scholar
(6)Gillis, J.Centrally biased discrete random walk. Quart. J. Math. (2), 7 (1956), 144–52.Google Scholar
(7)Good, I. J.Legendre polynomials and trinomial random walks. Proc. Camb. Phil. Soc. 54 (1958), 3942.CrossRefGoogle Scholar
(8)Harris, T. E.First passage and recurrence distributions. Trans. Amer. Math. Soc. 73 (1952), 471–83.CrossRefGoogle Scholar
(9)Joos, G.Theoretical physica (Glasgow, 1934).Google Scholar
(10)Kac, M.Random walk and the theory of Brownian motion. Amer. Math. Mon. 54 (1947), 369–91.CrossRefGoogle Scholar
(11)Pringsheim, A.Irrationalzahlen und Konvergenz unendlicher Prozesse. Enc. Math. Wiss. IA 3 (18981904), 47146.Google Scholar
(12)Uspensky, J. V.Introduction to mathematical probability (New York, 1937).Google Scholar
(13)Wall, H. S.Analytic theory of continued fractions (New York, 1948).Google Scholar
(14)Wang, M. C. and Uhlenbeck, G. E.On the theory of the Brownian motion. II. Rev. Mod. Phys. 17 (1945), 323–42.CrossRefGoogle Scholar