Asymptotic representations for hypergeometric-Bessel type function and fractional integrals

• Authors: Anatoly A. Kilbas;Luis Rodríguez;Juan J. Trujillo
• Venue: Journal of Computational and Applied Mathematics
• Year: 2002

The paper is devoted to the study of asymptotic relations for the function λγ,σ(β)(z)=β/Γ(γ+1-1/β) ∫1∞ (tβ - 1)γ-1/βtσ e-ztdt generalising 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 (Iα_f)(x)=1/Γ(α)∫x∞ f(t)dt/(t-x)1-α (x 0; α ∈ C, Re(α) 0) and for the Erdelyi-Kober-type fractional intergral (Iα_;β,ηf)(x)= βxβη/Γ(α) ∫x∞(tβ(1-α-η)-1/ f(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.