What is closed form of following series? $$t^{(\frac{b}{2}-1)}\sum_{n=0}^\infty\sum_{k=0}^\infty\frac{(i^{b})^{n}t^{b(n-\frac{k}{2})}z^{k}}{k!Γ(b(n-\frac{k}{2})+\frac{b}{2})}$$
I solved a problem with two different methods. With the first method, the answer is given as follows: $$e^{z}E_{b}(it),$$ where $E_{b}(it)$ is Mittag-Leffler function. With the second method, I obtained the above series. Is this series equal to the answer obtained with the first method?