NFA-$\epsilon$ extended transition function for inverted strings It is well known that in $NFA-\epsilon$ the extended transition function is defined as it follows:
\begin{align*}
\hat\delta: Q &amp;\times \Sigma^* \rightarrow \mathbb{P}(Q) \\
\hat\delta(q,\epsilon) &amp;= ECLOSURE(q) \\
\hat\delta(q,\alpha x) &amp;= \bigcup_{p_i \in \hat\delta(q,\alpha)}ECLOSURE(\delta(p_i,x)) 
\text{ for } \alpha \in \Sigma^* \text{ and } x \in \Sigma
\end{align*}

I would like to give an equivalent definition for the function but now considering an string $w \in \Sigma^* $ such that $w = x\alpha$ with the same definitions for $x$ and $\alpha$ given above. But what I've only got is the base case (i.e. $w = \epsilon$) which is pretty the same. Any hints to define the function like that?
First, there's a typo at your definition of $\hat\delta$. Case $\hat\delta(q,\alpha x)$ should be $\bigcup_{p_i \in \hat\delta(q,\alpha)}ECLOSURE(\delta(\color{green}{p_i},x))$ (green for highlighting the difference).

The equivalent definition in terms of a string given as $(x\alpha)$, where $x\in\Sigma$ and $\alpha\in\Sigma^*$, is
$$
\hat\delta(q,x\alpha) = \bigcup_{p_i \in ECLOSURE(\delta(q,x))} \hat\delta(p_i,\alpha)
$$