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On Boole's operational solution of linear finite difference equations with rational coefficients

Published online by Cambridge University Press:  24 October 2008

L. M. Milne-Thomson
Affiliation:
Corpus Christi College

Extract

The solution of linear differential equations by the method of Frobenius is a straightforward matter consisting merely in the substitution of a power series and obtaining the coefficients from a recurrence relation. The corresponding method when applied to difference equations is laborious and leads in general to divergent series from which convergent solutions can be obtained by a method due to Birkhoff(1). Actually a more appropriate method is to find a factorial series, but even here the direct substitution of such a series requires repeated transformations and throws little light on the structure of the equation. A method of symbolic operators was originated by Boole(2), but owing to the restricted definition of the operators it has a very limited scope. In the present paper a more general interpretation is given to the operators, and their application to a certain class of linear difference equations with rational coefficients is discussed.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

REFERENCES

(1)Birkhoff, G. D., Trans. Amer. Math. Soc., 12 (1911), 243284.CrossRefGoogle Scholar
(2)Boole, G., Calculus of Finite Differences, London (1872).Google Scholar
(3)Nörlund, N. E., Leçons sur les équations linéaires, Paris (1929).Google Scholar