MHB Has anybody seen this looks kind double Mittag-Leffler

[sarrah1]

Hi
I got stuck with this, it looks like a double mittag-leffler. Has anybody seen it

$$\sum_{k=0}^{\infty}\sum_{j=o}^{\infty} \frac{{t}^ {\alpha (j+k)} {a}^{k} c {b}^{j}} {\varGamma(\alpha j+\alpha k+\alpha+1)} $$

thanks

I wonder why the symbols look so small in the post. the numerator is t raised to alpha(j+k) times a^k times constant c times b^j
thanks
sarrah

 

[Jhoneine]

This is a double Mittag-Leffler function, which is defined as:$$M(t,a,b,\alpha) = \sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\frac{t^{\alpha(j+k)}a^kb^j}{\Gamma(\alpha j+\alpha k+\alpha+1)}.$$This function is used to represent solutions of certain fractional differential equations. It is a generalization of the classical Mittag-Leffler function.