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Generalized fractional integration of Bessel function of the first kind. (English) Zbl 1156.26004

The authors study two integral transforms involving the \({}_2F_1\) hypergeometric function. These integral transforms are generalizations of both Riemann-Liouville fractional integrals and Erdélyi-Kober fractional integrals. The integral transforms are applied to the Bessel function of the first kind of order \(\nu\). The special cases \(\nu=-1/2\) and \(\nu=1/2\) finally lead to fractional integrals involving the cosine and the sine function, respectively.

MSC:

26A33 Fractional derivatives and integrals
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
33C20 Generalized hypergeometric series, \({}_pF_q\)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
26A09 Elementary functions
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References:

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[2] Erdélyi A., Higher Transcendental Functions 2 (1953) · Zbl 0051.30303
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[4] DOI: 10.1112/plms/s2-27.1.389 · JFM 54.0392.03
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[10] DOI: 10.1098/rsta.1940.0002 · Zbl 0023.14002
[11] DOI: 10.1112/plms/s2-46.1.389 · Zbl 0025.40402
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