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On an index law and a result of Buschman

Published online by Cambridge University Press:  14 November 2011

Adam C. McBride
Adam C. McBride
Affiliation:
Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G l 1XH
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Synopsis

A result for the Erdélyi-Kober operators, mentioned briefly by Buschman, is discussed together with a second related result. The results are proved rigorously by means of an index law for powers of certain differential operators and are shown to be valid under conditions of great generality. Mellin multipliers are used and it is shown that, in a certain sense, the index law approach is equivalent to, but independent of, the duplication formula for the gamma function. Various statements can be made concerning fractional integrals and derivatives which produce, as special cases, simple instances of the chain rule for differentiation and changes of variables in integrals.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 96 , Issue 3-4 , 1984 , pp. 231 - 247
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

1Buschman, R. G.. Fractional integration. Math. Japon. 9 (1964), 99–106.Google Scholar
2Erdélyi, A. et al. Higher transcendental functions, Vol. 1 (New York: McGraw-Hill, 1953).Google Scholar
3McBride, A. C.. Fractional calculus and integral transforms of generalized functions. Lecture Notes in Mathematics 31 (London: Pitman, 1979).Google Scholar
4McBride, A. C.. Fractional powers of a class of ordinary differential operators. Proc. London Math. Soc. 45 (1982), 519–546.CrossRefGoogle Scholar
5McBride, A. C.. Fractional powers of a class of Mellin multiplier transforms I. Submitted.Google Scholar
6McBride, A. C.. Fractional powers of a class of Mellin multiplier transforms II. To appear in Applicable Analysis.Google Scholar