Asymptotic representations for hypergeometric-Bessel type function and fractional integrals

The paper is devoted to the study of asymptotic relations for the functionλγ,σ(β)(z)=βΓ(γ+1−1/β)∫1∞(tβ−1)γ−1/βtσ e−zt dtgeneralising Tricomi confluent hypergeometric function and modified Bessel function of the third kind. The full asymptotic representations for λγ,σ(β)(z) at zero and infinity are established. Applications are given to obtain full asymptotic expansions near zero and infinity for the Liouville fractional integral(I−αf)(x)=1Γ(α)∫x∞f(t) dt(t−x)1−α (x>0; α∈C, Re(α)>0)and for the Erdelyi–Kober-type fractional integral(I−;β,ηαf)(x)=βxβηΓ(α)∫x∞tβ(1−α−η)−1f(t) dt(tβ−xβ)1−α (x>0; α∈C, (Re(α)>0)with β>0 and η∈C of power-exponential function f(t), and for three other fractional integrals.