Do there exist continuous functions $c_n\colon\mathbb R^+\to\mathbb R$ and $\gamma\colon\mathbb R^+\to\mathbb R$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$ and the following equation is true for all $\alpha,x>0$ $$ e^{-\alpha x}=\sum_{n=1}^{\infty}c_n\left(\alpha\right)\gamma\left(x\right)^n $$ If yes, how regular can we make $\gamma$, can it be smooth? Analytic? What about the $c_n$?

**Clarifying edit:** all the $x$-dependence is in $\gamma$ and all the $\alpha$ dependence is in the $c_n$