A modified Bessel-type integral transform and its compositions with fractional calculus operators on spaces Fp,μ and Fp,μ′

The paper is devoted to study the integral transform(Lγ,σ(β)f)(x)=∫0∞λγ,σ(β)(xt)f(t) dt (x>0)with the kernelλγ,σ(β)(z)=βΓ(γ+1−1/β)∫1∞(tβ−1)γ−1/βtσe−zt dtfor β>0; Re(γ)>1/β−1; σ∈R; Re(z)>0, which is a generalization of the modified Bessel function of the third kind or Macdonald function K−γ(z). Properties of λγ,σ(β)(z) are investigated and compositions of the operator Lγ,σ(β) with the left- and right-sided Liouville fractional integrals and derivatives are proved.