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A theorem on a finite difference operator and its connection with the Poisson distribution
Published online by Cambridge University Press: 09 April 2009
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The well-known Taylor expansion of a function around a point a can be formally written as 
The last expression is just a symbolic form and is valid, as we know, under certain restrictive conditons. The last expression is just a symbolic form and is valid, as we know, under certain restrictive conditons. We shall study the situation when the differential operator d/da is replaced by the finite difference operator Δh/h, where the operator Δh, is defined by 
In general, 
Then we have the following theorem.
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- Copyright © Australian Mathematical Society 1966
References
[1]Bernstein, S., “Démonstration du théoréme de Weierstrass fondée sur le calcul des probabilités”, Communic. Soc. Math. Cracow, vol. 13, p. 1.Google Scholar
[2]Cramér, H., Mathematical methods of statistics, Princeton University Press, Princeton.Google Scholar
[3]Hille, E. and Phillips, R. S., Functional analysis and semi-groups, Amer. Math. Soc. Colloquium Publications, vol. XXXI, 1957.Google Scholar
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