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A hypergeometric integral equation

Part of the Lecture Notes in Mathematics book series (LNM,volume 457)

Abstract

The aim is to find conditions under which the equation

$$\int_0^\infty {F(a,b;c; - \tfrac{x}{t})t^{ - b} f(t)dt = g(x)\;\,\,\,\,\,for\,\,\,0 < x < \infty } $$

has a solution for f; F being the hypergeometric function. The method uses fractional differentiation to reduce the left side first to a generalized Stieltjes transform and then to a Stieltjes transform; the latter is then inverted by an adaptation of one of Widder's methods. The results obtained are: necessary conditions for a solution in a certain class, sufficient conditions for a solution in a comparable class, and an explicit formula for the solution, involving fractional derivatives.

Keywords

  • Fractional Derivative
  • Hypergeometric Function
  • Fractional Integral
  • Fractional Differentiation
  • Absolute Convergence

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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© 1975 Springer-Verlag

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Love, E.R. (1975). A hypergeometric integral equation. In: Ross, B. (eds) Fractional Calculus and Its Applications. Lecture Notes in Mathematics, vol 457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067112

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  • DOI: https://doi.org/10.1007/BFb0067112

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-07161-7

  • Online ISBN: 978-3-540-69975-0

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