If I solve the Airy differential equation ($ {y'' - xy = 0}$) using the power series method, I get the following solution:
$y(x) = a_0 \left(1 + \frac{x^3}{6} + \frac{x^6}{180} + \cdots \right) + a_1 \left(x + \frac{x^4}{12} + \frac{x^7}{504} + \cdots \right)$
However, everywhere I research the Airy differential equation, I see answers in terms of $Ai(x)$ and $Bi(x)$, which are expressed using gamma functions and fractional powers of 3.
My question is how do we get from the series solution (where I am at) to $Ai(x)$ and $Bi(x)$?