Mellin transforms of generalized fractional integrals and derivatives

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 delta(r,m) operators with generalized Stirling numbers and Lah numbers. For example, we show that delta(1,1) corresponds to the Stirling numbers of the 2nd kind and delta(2,1) corresponds to the unsigned Lah numbers. Further, we show that the two operators delta(r,m) and delta(m,r), r, m is an element of N, generate the same sequence given by the recurrence relation. S(n, k) = Sigma(r)(i=0) (m +(m - r)(n - 2) + k - i - 1)(r-i) ((r)(i))S(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 is an element of {1/3, 1/4, 1/5, 1/6, ... } and in turn show that delta(1/2,1) corresponds to the generalized Laguerre polynomials. (C) 2014 Elsevier Inc. All rights reserved.