Airy coefficient recurrence and geometric tail bounds

Multiplying the coefficient relation in [F1] by $z^{k+3}$ shows that the coefficient of $z^{k+1}$ in $y^{\mathrm{\prime }\mathrm{\prime }}$ is $c_k$, which is the coefficient of $z^{k+1}$ in $zy$. At degree $0$, the coefficients agree because $c_2=0$; at degree $1$, they agree by the recurrence at $k=0$. Thus the formal series satisfy $y^{\mathrm{\prime }\mathrm{\prime }}=zy$.

For $T_k=c_k{z}^k$ with $c_k\mathrm{\ne }0$, $\mathrm{\mid }{T}_{k+3}\mathrm{\mid }\mathrm{/}\mathrm{\mid }{T}_k\mathrm{\mid }=\mathrm{\mid }z{\mathrm{\mid }}^3\mathrm{/}((k+3)(k+2))$. This tends to zero and decreases in $k$; once all subsequent ratios are at most $q<1$, summing the geometric majorant gives ${\sum }_{j\ge 0}\mathrm{\mid }{T}_{k_0+3j}\mathrm{\mid }\le \mathrm{\mid }{T}_{k_0}\mathrm{\mid }\mathrm{/}(1-q)$.

On every disk $\mathrm{\mid }z\mathrm{\mid }\le R$, the ratios of each residue-class term and of its first two formal derivative series tend to zero uniformly as $k\to \mathrm{\infty }$. Applying the geometric bound from step 1.2 to each class gives locally uniform convergence of the series and differentiated series. The series have infinite radius by these bounds, so [F2] permits termwise differentiation; step 1.1 then proves the Airy equation for the sums.