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On Certain Functional Identities in EN

Published online by Cambridge University Press:  20 November 2018

A. McD. Mercer
A. McD. Mercer*
Affiliation:
University of Guelph, Guelph, Ontario
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1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity

1.1

in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to

1.2

In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of

1.3

to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 315 - 324
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Flanders, H., Differential forms with applications to the physical sciences (Academic Press, New York-London, 1963).Google Scholar
2. Landau, E., Vorlesungen ùber Zahlen théorie, Vol. II (Chelsea, New York, 1946-47).Google Scholar
3. Mercer, A. McD., A uniqueness theorem for a class of Bess el function expansions, Quart. J. Math. Oxford Ser. 21 (1970), 83–87.Google Scholar