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Mathematics of Computation

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Exponential laws for fractional differences

Author: Godfrey L. Isaacs
Journal: Math. Comp. 35 (1980), 933-936
MSC: Primary 39A05
MathSciNet review: 572866
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Abstract: In Math. Comp., v. 28, 1974, pp. 185-202, Diaz and Osler gave the following (formal) definition for $ {\dot \Delta ^\alpha }f(z)$, the $ \alpha $th fractional difference of $ f(z):{\dot \Delta ^\alpha }f(z) = \Sigma _{p = 0}^\infty A_p^{ - \alpha - 1}f(z + \alpha - p)$. They derived formulas and applications involving this difference. They asked whether their differences satisfied an exponent law and what the relation was between their differences and others, such as $ {\Delta ^\alpha }f(z) = \Sigma _{p = 0}^\infty A_p^{ - \alpha - 1}f(z + p)$. In this paper an exponent law for their differences is established and a relation found between the two differences mentioned above. Applications of these results are given.

References [Enhancements On Off] (What's this?)

  • [1] S. CHAPMAN, "On non-integral orders of summability of series and integrals," Proc. London Math. Soc. (2), v. 9, 1911, pp. 369-409.
  • [2] J. B. DIAZ & T. J. OSLER, "Differences of fractional order," Math. Comp., v. 28, 1974, pp. 185-202. MR 0346352 (49:11077)
  • [3] G. L. ISAACS, "An iteration formula for fractional differences," Proc. London Math. Soc. (3), v. 13, 1963, pp. 430-460. MR 0155121 (27:5061)

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Keywords: Fractional differences, successive differences, exponent law, summability of series
Article copyright: © Copyright 1980 American Mathematical Society

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