## 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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