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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.

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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.

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    $\begingroup$ Isn't the MLF with $\alpha=\beta=1$ the exponential function $e^z$? $\endgroup$
    – gciriani
    Commented Mar 15, 2023 at 22:22
  • $\begingroup$ @gciriani Yes, it is $\endgroup$
    – Quiet_waters
    Commented Apr 4, 2023 at 18:32

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