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.


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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