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On an index law and a result of Buschman

  • Adam C. McBride (a1)

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.

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1Buschman, R. G.. Fractional integration. Math. Japon. 9 (1964), 99106.
2Erdélyi, A. et al. Higher transcendental functions, Vol. 1 (New York: McGraw-Hill, 1953).
3McBride, A. C.. Fractional calculus and integral transforms of generalized functions. Lecture Notes in Mathematics 31 (London: Pitman, 1979).
4McBride, A. C.. Fractional powers of a class of ordinary differential operators. Proc. London Math. Soc. 45 (1982), 519546.
5McBride, A. C.. Fractional powers of a class of Mellin multiplier transforms I. Submitted.
6McBride, A. C.. Fractional powers of a class of Mellin multiplier transforms II. To appear in Applicable Analysis.

On an index law and a result of Buschman

  • Adam C. McBride (a1)

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