On the Laguerre fractional integro-differentiation. (English) Zbl 1476.44004
This paper deals with the constructions of the fractional integrodifferentiation, which represent fractional powers of the so-called Laguerre derivative \(\theta\equiv DxD\), where \(D\equiv d/dx\) is the differential operator [G. Dattoli and A. Torre, Atti Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 132, 3–9 (1998; Zbl 1098.33501); G. Dattoli et al., Integral Transforms Spec. Funct. 15, No. 4, 309–321 (2004; Zbl 1056.35036)]. The main feature of the Laguerre derivative is that its integer powers satisfy a Viskov-type identity [O. V. Viskov and H. M. Srivastava, J. Math. Anal. Appl. 186, No. 1, 1–10 (1994; Zbl 0802.33008)]. The author investigates fractional analogs of the Laguerre operators when the positive integer \(n\) is replaced by the real positive \(\alpha\). Their mapping properties in spaces \(L_{v,p}(\mathbb{R}_+)\), semigroup properties and formulas of the integration by parts as well as their Mellin-Barnes representations are studied. Further, a Volterra integral equation of second kind involving the Laguerre fractional integral is solved in terms of the double hypergeometric type series as the resolvent kernel.
Reviewer: S. L. Kalla (Ballwin)
MSC:
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 26A33 | Fractional derivatives and integrals |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
Keywords:
fractional integro-differentiation; Laguerre derivative; Mellin transform; Gauss hypergeometric function; double hypergeometric seriesReferences:
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