4. Given that $\left\{a_{n}\right\}$ is an arithmetic sequence with a non-zero common difference, $\left\{b_{n}\right\}$ is a geometric sequence, where $a_{1}=3, b_{1}=1, a_{2}=b_{2}$, and for any number $n$, $a_{n}=\log _{\alpha} b_{n}+\beta$. Then $\alpha+\beta=$ $\qquad$ .
4. $\sqrt[3]{3}+3$

Let $\left\{a_{n}\right\}$ have a common difference of $d$, and $\left\{b_{n}\right\}$ have a common ratio of $q$. Then
$$
\left\{\begin{array}{l}
3+d=q, \\
3(3+4 d)=q^{2} .
\end{array}\right.
$$

Solving, we get $d=6, q=9$.
Thus, $3+6(n-1)=\log _{\alpha} 9^{n-1}+\beta$ holds for all
positive integers $n$.
Therefore, $\log _{\alpha} 9=6, \beta=3$.

Solving, we get $\alpha=\sqrt[3]{3}, \beta=3 \Rightarrow \alpha+\beta=\sqrt[3]{3}+3$.