I am studying properties of the **two-parameter** Mittag-Leffler function.
$$ E\_{\alpha,\beta}(z)=\sum\_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}.$$
I am particularly interested in recurrences and relations, such as the duplication formulas.
However, I am seeking some relation in which **a product of two two-parameter Mittag-Leffler functions is a Mittag-Leffler function**, does anyone know something about this? Maybe about powers, but not necessarily.



When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function?
The product of two MLF is another MLF, only when $\alpha = \beta =1.$ In the other cases we obtain a similar formula but have to introduce generalized binomial coefficients. To show it, it is enough to use the rule used for multiplication of power series. The coefficients of the product are the discrete convolution of the coefficients of both series.


