= Airy power-series fundamental pair
{c}
{title2=$y_1(0)=1,\ y_1'(0)=0,\ y_2(0)=0,\ y_2'(0)=1$}
For the <Airy ordinary differential equation> $y''=xy$, a <power series> obeys $a_2=0$ and $a_{m+3}=a_m/[(m+3)(m+2)]$. Setting its first two coefficients to $(1,0)$ or $(0,1)$ gives
$$
y_1(x)=\sum_{k\geq0}\frac{x^{3k}}{\prod_{j=1}^k(3j)(3j-1)},\qquad
y_2(x)=\sum_{k\geq0}\frac{x^{3k+1}}{\prod_{j=1}^k(3j)(3j+1)}.
$$
Empty products are one. The <ratio test> proves convergence on the whole complex plane. Termwise differentiation verifies the equation, and the initial normalization gives <Wronskian> one. By the <Abel identity>, the <Wronskian> remains one everywhere, so these solutions span the whole solution space.
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