Mellin Transforms of the 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 ${\delta }_{r,m}$ operators with generalized Stirling numbers and Lah numbers. For example, we show that ${\delta }_{1,1}$ corresponds to the Stirling numbers of the $2^{nd}$ kind and ${\delta }_{2,1}$ corresponds to the unsigned Lah numbers. Further, we show that the two operators ${\delta }_{r,m}$ and ${\delta }_{m,r}$, $r,m\in \mathbb{N}$, generate the same sequence given by the recurrence relation \[ S(n,k)=\sum_{i=0}^r \big(m+(m-r)(n-2)+k-i-1\big)_{r-i}\binom{r}{i} S(n-1,k-i), \;\; 0< k\leq n, \] 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 $k\le 0$. Finally, we define a new class of sequences for $r\in \{\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},...\}$ and in turn show that ${\delta }_{\frac{1}{2},1}$ corresponds to the generalized Laguerre polynomials.

17 pages, 1 figure, 9 tables, Accepted for publication in Applied Mathematics and Computation

References in corpus (2)

• Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative
• Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

Cited by in corpus (4)

• Hermite-Hadamard and Hermite-Hadamard-Fejér type Inequalities for Generalized Fractional Integrals
• Existence and Uniqueness results for a class of Generalized Fractional Differential Equations
• Theory of Nonlinear Caputo-Katugampola Fractional Differential Equations
• A Systematic Review on Hermite-Hadamard Inequality: Theory and Applications