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

The paper is devoted to study the integral transform (L(gamma,sigma)((beta))f)(x) = integral(0)(infinity)lambda(gamma sigma)((beta))(xt)f(t)dt (x>0)with the kernel lambda(gamma,sigma)((beta))(z)=beta/Gamma(gamma+1-1/beta)integral(1)(infinity)(t(beta)-1)(gamma-1/beta)t(sigma)e(-zt) dt for beta>0; Re(gamma)>1/beta-1; sigma element of R; Re(z)>0, which is a generalization of the modified Bessel function of the third kind of Macdonald function K-gamma(z). Properties of lambda(gamma,sigma)((beta))(z) investigated and compositions of the operator L-gamma,sigma((beta)) with the left- and right-sided Liouville fractional integrals and derivatives are proved. (C) 2000 Elsevier Science B.V. All rights reserved.