DSolve[(z^2)D[f[z],{z,2}]-(dim-1)z D[f[z],z]-(k^2)(z^2)f[z]==(mass^2) f[z],f[z],z]//FullSimplify

Assuming[{Δ>(dim/2),dim>0,k>0,z>0},Series[(2/(Gamma[Δ-(dim/2)]))((((2/k)))^(dim/2 - Δ))(z^(dim/2))BesselK[Δ-(dim/2),k z],{z,0,0}]//FullSimplify]

(* The first term vanishes as z->0 since Δ>0, the second gives (z^(dim-Δ)) *)

;

Integrate[(α^(λ-1))(E^(-α-((x^2)/(4α)))),{α,0,∞}]

BesselK[k,x]-BesselK[-k,x]//FullSimplify