Can the following series be written as a Mittag -Leffler function?
$$\sum_{n=0}^\infty\frac{(z)^{(2n+1)}}{Γ((2n+1)\alpha+\alpha+1)}-\sum_{n=0}^\infty\frac{(z)^{(2n)}}{Γ(2n\alpha+\alpha+1)}$$ we have for one parameter Mittag-Leffler function : $$cos_{\alpha}(t^{\alpha})=\sum_{k=0}^\infty(-1)^{k}\frac{t^{2\alpha k}}{Γ(2\alpha k+1)}$$ $$sin_{\alpha}(t^{\alpha})=\sum_{k=0}^\infty(-1)^{k}\frac{t^{(2k+1)\alpha}}{Γ(2\alpha k+\alpha+1)}$$ Can I write for the given series in terms of two parameters fractional trigonometric function ? for example in terms of $E_{\alpha,\alpha+1}$?
We can simplify the expression as the following: $$ z\cdot\sum_{n=0}^\infty\frac{z^{2n}}{\Gamma(2\alpha n+2\alpha+1)}-\sum_{n=0}^\infty\frac{z^{2n}}{\Gamma(2\alpha n+\alpha+1)} = \\ z\cdot E_{2\alpha, 2\alpha+1}(z^2) - E_{2\alpha,\alpha+1}(z^2), $$ where $$ E_{\alpha,\beta}(z)=\sum_{n=0}^{\infty}\frac{z^n}{\Gamma(\alpha n+\beta)} $$ is the generalized Mittag-Leffler's function, with $z\in\mathbb{C}$ and $\alpha>0$.
You can find more details in my two related answers here, and here.