I find the following identity in many special functions books without proof. This identity is called the Laplace transform of the Mittag-Leffler function with three parameters. The result is in the form of Fox-Wright function.

$$\int _0^{\infty }e^{-q t}t^{\rho -1}E_{\alpha ,\theta }^{\gamma }\left(w t^{\beta }\right)dt=\frac{q^{-\rho }}{\Gamma(\gamma )}\, _2\psi _1\left[\frac{w}{q^{\beta }}| \begin{array}{cc} (\gamma ,1) & (\rho ,\beta ) \\ (\theta ,\alpha ) & \\ \end{array} \right],$$ where $\Re(\alpha )$, $\Re(\theta )$, $\Re(\gamma )$,$\Re(\rho )$,$\Re(q)>0$ and $q>|w|^{\frac{1}{\Re(\alpha)}}$.

This identity is taken from $Eq.~(5.1 .30)$ in "Mittag - Leffler Functions, Related Topics and Applications", Springer. The series expression of three parameter Mittag-Leffler function is $$E_{\alpha ,\theta }^{\gamma }(z)=\sum _{k=0}^{\infty } \frac{(\gamma )_k}{k!\Gamma(\alpha k+\theta )}z^k,\alpha ,\theta ,\gamma >0.$$ You may other representation of this function in the same chapter. Definitions and convergence of the Fox-Wright function are in Appendix F of that book.

I suppose the proof is expanding the Mittag-Leffler function into series and then integrating term by term. Therefore, the last condition is for the uniformly convergence of the Fox-Wright function. However, what if $q\leq|w|^{\frac{1}{\Re(\alpha)}}$. Do we have any other way to deal with this case?

  • $\begingroup$ It would be useful if you gave some references and the definitions of the three-parameter Mittag-Leffler function and the Fox-Wright function. Have you tried to derive this Laplace transform? $\endgroup$ – Pantelis Sopasakis May 13 at 0:53
  • $\begingroup$ Sure thing. I have modified my question.However, the series expansion might be restricted to the convergence issue and therefore I did not write down the Fox-Wright function. $\endgroup$ – gouwangzhangdong May 14 at 1:12
  • $\begingroup$ My guess is that the RHS is a meromorphic function of $q$ defined by a contour integral and we know its poles and decay, enough to apply the residue theorem to its inverse Laplace transform integral and obtain $t^{\rho-1}$ times the power series of $t^\beta$ you mentioned (see en.wikipedia.org/wiki/Hypergeometric_function#Barnes_integral and en.wikipedia.org/wiki/Fox_H-function) $\endgroup$ – reuns May 14 at 1:36

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