Mellin transforms of generalized fractional integrals and derivatives

Abstract

We obtain the Mellin transforms of the generalized fractional integrals and derivatives that generalize the Riemann–Liouville and the Hadamard fractional integrals and derivatives. We also obtain interesting results, which combine generalized δr,m operators with generalized Stirling numbers and Lah numbers. For example, we show that δ1,1 corresponds to the Stirling numbers of the 2nd kind and δ2,1 corresponds to the unsigned Lah numbers. Further, we show that the two operators δr,m and δm,r,r,m∈N, generate the same sequence given by the recurrence relation.S(n,k)=∑i=0rm+(m-r)(n-2)+k-i-1r-iriS(n-1,k-i),0<k⩽n,with S(0,0)=1 and S(n,0)=S(n,k)=0 for n>0 and 1+min{r,m}(n-1)<k or k⩽0. Finally, we define a new class of sequences for r∈13,14,15,16,… and in turn show that δ12,1 corresponds to the generalized Laguerre polynomials.