Erdelyi–Kober operator

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In mathematics, an Erdélyi–Kober operator is a fractional integration operation introduced by Arthur Erdélyi (1940) and Hermann Kober (1940).

The Erdélyi–Kober fractional integral is given by

${\displaystyle {\frac {x^{-\nu -\alpha +1}}{\Gamma (\alpha )}}\int _{0}^{x}(t-x)^{\alpha -1}t^{-\alpha -\nu }f(t)dt}$

which generalizes the Riemann fractional integral and the Weyl integral.

Comparison

There is a similar operator now known as the Katugampola fractional operator which generalizes both the Riemann-Liouville and the Hadamard fractional integrals into a unique form.

References

• Erdélyi, A. (1940), "On fractional integration and its application to the theory of Hankel transforms", The Quarterly Journal of Mathematics. Oxford. Second Series, 11: 293–303, ISSN 0033-5606, MR 0003271, doi:10.1093/qmath/os-11.1.293
• Erdélyi, Arthur (1950–51), "On some functional transformations", Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino, 10: 217–234, MR 0047818
• Erdélyi, A.; Kober, H. (1940), "Some remarks on Hankel transforms", The Quarterly Journal of Mathematics. Oxford. Second Series, 11: 212–221, ISSN 0033-5606, MR 0003270, doi:10.1093/qmath/os-11.1.212
• Kober, Hermann (1940), "On fractional integrals and derivatives", The Quarterly Journal of Mathematics (Oxford Series), 11 (1): 193–211, doi:10.1093/qmath/os-11.1.193
• Sneddon, Ian Naismith (1975), "The use in mathematical physics of Erdélyi-Kober operators and of some of their generalizations", in Ross, Bertram, Fractional calculus and its applications (Proc. Internat. Conf., Univ. New Haven, West Haven, Conn., 1974), Lecture Notes in Math., 457, Berlin, New York: Springer-Verlag, pp. 37–79, ISBN 978-3-540-07161-7, MR 0487301, doi:10.1007/BFb0067097