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 $$y(x,\lambda)=|x|^{\omega/2}E_1(ix\lambda;1+\omega/2),\quad\omegaın(-1,1),$$$E_1(z;\alpha)=\sum\limits_{k=0}^{ınfty}\dfrac{z^k}{\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.