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 $k\leq 0$. 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.