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.
This is a repost of this question: https://mathoverflow.net/questions/402681/when-is-a-product-of-two-two-parameter-mittag-leffler-functions-a-mittag-leffler.
Thank you in advance!