2011/12/31 by Udita N. Katugampola · 1 citation
Mathematics · #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #math.CA #math.CO #msc:26A33 #msc:44A15 #msc:65R10
paper · pdf · doi:10.1016/j.amc.2014.12.067
published as Applied Mathematics and Computation 257 (2015) 566-580 · 17 pages, 1 figure, 9 tables, Accepted for publication in Applied Mathematics and Computation
arxiv created 2014/10/29 · openalex publication_date 2015/02/07 · arxiv updated 2015/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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=0r (m+(m-r)(n-2)+k-i-1)r-i\binomri 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 ∈ \(1)/(3), (1)/(4), (1)/(5), (1)/(6), ...\ and in turn show that δ(1)/(2),1 corresponds to the generalized Laguerre polynomials.