A basis property of families of the Mittag-Leffler functions

Abstract

In this paper we study basis properties of the Mittag-Leffler functions $\{y(x,{\lambda }_k)\}$ where $$y(x,\lambda )=|x{|}^{\omega /2}{E}_1(ix\lambda ;1+\omega /2),\omega ın(-1,1),$$$E_1(z;\alpha )=\sum _{k=0}^{ınfty}{\displaystyle \frac{z^k}{\mathrm{\Gamma }(\alpha +k)}}$ and $\{{\lambda }_k\}$ is a sequence of complex numbers. In the case $\omega =0$ this system is reduced to the exponential system.