Computation of fractional integrals via functions of hypergeometric and Bessel type (Q1571021)

The function \[ \lambda_{\gamma , \sigma }^{(\beta)}(z)={\beta \over \Gamma (\gamma +1-1/\beta)} \int_1^{\infty} (t^\beta -1)^{\gamma -1/\beta } t^\sigma e^{-zt} dt \] is considered for \(\beta >0\), \(\gamma , \sigma , z \in \mathbb{C}\), such that \(\Re (\gamma) +1-1/\beta >0\) and \(\Re (z)>0\). The authors discuss the special cases when \(\lambda_{\gamma , \sigma }^{(\beta)}\) can be expressed as a function of hypergeometric type (namely, the Tricomi confluent hypergeometric function) or in terms of the modified Bessel function of the third kind. For \(z >0\) these functions are representable as fractional integrals. Finally, a connection of the right-sided Liouville fractional integral with the function \(\lambda_{\gamma , \sigma }^{(\beta)}\) allows the computation of the former. Other fractional integrals of power exponential functions are also considered.