Kilbas, Anatoly A.; Sebastian, Nicy Generalized fractional integration of Bessel function of the first kind. (English) Zbl 1156.26004 Integral Transforms Spec. Funct. 19, No. 12, 869-883 (2008). 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. Reviewer: Roelof Koekoek (Delft) Cited in 32 Documents 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 Keywords:fractional integral transforms; Bessel function of the first kind; hypergeometric function; generalized hypergeometric function PDFBibTeX XMLCite \textit{A. A. Kilbas} and \textit{N. Sebastian}, Integral Transforms Spec. Funct. 19, No. 12, 869--883 (2008; Zbl 1156.26004) Full Text: DOI References: [1] Erdélyi A., Higher Transcendental Functions 1 (1953) · Zbl 0051.30303 [2] Erdélyi A., Higher Transcendental Functions 2 (1953) · Zbl 0051.30303 [3] Erdélyi A., Higher Transcendental Functions 3 (1954) · Zbl 0143.29202 [4] DOI: 10.1112/plms/s2-27.1.389 · JFM 54.0392.03 · doi:10.1112/plms/s2-27.1.389 [5] DOI: 10.1201/9780203487372 · doi:10.1201/9780203487372 [6] Kilbas A. A., Fract. Calc. Appl. Anal. 54 pp 437– (2002) [7] Saigo M., Math. Rep. Kyushu Univ. 11 pp 135– (1978) [8] Samko S. G., Fractional Integrals and Derivatives. Theory and Applications (1993) [9] DOI: 10.1112/jlms/s1-10.40.286 · Zbl 0013.02104 · doi:10.1112/jlms/s1-10.40.286 [10] DOI: 10.1098/rsta.1940.0002 · Zbl 0023.14002 · doi:10.1098/rsta.1940.0002 [11] DOI: 10.1112/plms/s2-46.1.389 · Zbl 0025.40402 · doi:10.1112/plms/s2-46.1.389 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.