On a representation of the \(E_\rho(z,\mu)\) function (Q1382918)

The authors prove the following integral representation: \[ E_\rho(z,\mu)=\int_0^\infty K_{1/\rho}(\rho^{-1},\mu,zx)x^{-\rho}g(x^{-\rho},\rho^{-1})dx,\quad z\in\mathbb C, \mu>\rho^{-1}, \rho>1, \] where \[ \begin{aligned} E_\rho(z,\mu)&:=\sum_{k=0}^\infty z^{k}/\Gamma(\mu+k/\rho) \text{ is a Mittag--Leffler type entire function,} \\ K_{c}(a,b,z)&:=\sum_{k=1}^\infty \frac{\Gamma(a+kc)}{\Gamma(b+kc)}\frac{z^{k}}{k!} \text{ is a hypergeometric function, and}\\ g(x,\alpha)&:=\frac 1\pi\sum_{k=1}^\infty(-1)^{k} \frac{\Gamma(k\alpha+1)}{\Gamma(k+1)}\sin(\pi k\alpha)x^{-k\alpha-1} \text{ is a probability density.}\end{aligned} \] .