Computation of fractional integrals via functions of hypergeometric and Bessel type

The paper is devoted to computation of the fractional integrals of power exponential functions. It is considered a function λγ,σ(β)(z) defined byλγ,σ(β)(z)=βΓ(γ+1−1/β)∫1∞(tβ−1)γ−1/βtσe−zt dtwith positive β and complex γ, σ and z such that Re(γ)>(1/β)−1 and Re(z)>0. The special cases are discussed when λγ,σ(β)(z) is expressed in terms of the Tricomi confluent hypergeometric function Ψ(a,c;x) and of modified Bessel function of the third kind Kγ(x). Representations of these functions via fractional integrals are proved. The results obtained apply to compute fractional integrals of power exponential functions in terms of λγ,σ(β)(x), Ψ(a,c;x) and Kγ(x). Examples are considered.