Parameter-uniform convergence of formal Euler series

Fix compact $K\subset B$ and $0<\rho <R<q$. Taylor-expand $f_k(b,x)={\sum }_{t\ge 0}{a}_{k,t}(b)x^t$ and $g(b,x)={\sum }_{t\ge 0}{b}_t(b)x^t$. On $K\times \{\mathrm{\mid }x\mathrm{\mid }=R\}$ joint continuity gives one bound $C\ge 1$ for all these finitely many functions. The Cauchy coefficient integral therefore gives $\mathrm{\mid }{a}_{k,t}(b)\mathrm{\mid }\le C{R}^{-t}$ and $\mathrm{\mid }{b}_t(b)\mathrm{\mid }\le C{R}^{-t}$, uniformly for $b\in K$. The same fixed-circle integral shows that all coefficient functions are continuous in $b$. Define $P_0(n,b)=n^N+{\sum }_{k=0}^{N-1}{a}_{k,0}(b)n^k$ and $P_t(l,b)={\sum }_{k=0}^{N-1}{a}_{k,t}(b)l^k$ for $t\ge 1$. Coefficient comparison in the formal equation, using $0^0=1$ for the identity operator, yields $P_0(n,b)c_n(b)=b_n(b)-{\sum }_{l=0}^{n-1}{P}_{n-l}(l,b)c_l(b)\mathrm{.}$ For $t\ge 1$, $\mathrm{\mid }{P}_t(l,b)\mathrm{\mid }\le NC{R}^{-t}(1+l{)}^{N-1}$, including $l=0$.

For every $A>0$ there is an integer $n_0\ge 1$ such that $\mathrm{\mid }{P}_0(n,b)\mathrm{\mid }\ge AC(1+n{)}^{N-1}$ for every $b\in K$ and $n\ge n_0$. Indeed $\mathrm{\mid }{P}_0(n,b)\mathrm{\mid }\mathrm{/}{n}^N\ge 1-C{\sum }_{k=0}^{N-1}{n}^{k-N}$, which is at least $1\mathrm{/}2$ for sufficiently large $n$; increasing $n_0$ then gives $n^N\mathrm{/}2\ge AC(1+n{)}^{N-1}$. Division is used only at these large degrees, never at possibly resonant small degrees. Put $d_n=R^n{\sup }_K\mathrm{\mid }{c}_n\mathrm{\mid }$, finite by continuity. The recurrence and $(1+l{)}^{N-1}\le (1+n{)}^{N-1}$ for $l<n$ give $d_n\le A^{-1}(1+N{\sum }_{l<n}{d}_l)(n\ge n_0)\mathrm{.}$ Choose finite nonnegative $D_n\ge d_n$ for $n<n_0$, and define $D_n=A^{-1}(1+N{\sum }_{l<n}{D}_l)$ thereafter. Induction on $n$ gives $d_n\le D_n$. For $n\ge n_0$ the defining recurrence gives $D_{n+1}=(1+N\mathrm{/}A)D_n$. Hence ${\sum }_n{D}_n{s}^n<\mathrm{\infty }$ for $0\le s<(1+N\mathrm{/}A{)}^{-1}$. Choose $A$ large enough that $\rho \mathrm{/}R<(1+N\mathrm{/}A{)}^{-1}$. Then ${\sum }_n{\sup }_K\mathrm{\mid }{c}_n\mathrm{\mid }{\rho }^n={\sum }_n{d}_n(\rho \mathrm{/}R{)}^n<\mathrm{\infty }$.

Every point of $B$ has a compact neighborhood contained in $B$, so the estimate proves local uniform convergence of the continuous partial sums, giving continuity of the sum. It also proves holomorphy in $x$: on a small disk, uniform convergence allows the sums to pass through the Cauchy integrals, giving the Cauchy formula for the sum. The Cauchy derivative formula on a slightly larger circle gives locally uniform convergence of all termwise $x$ derivatives. If each $c_n$ is holomorphic in $b$, choose a compact parameter polydisc and an $x$ disk with slightly larger concentric polydiscs still in the domain. The preceding uniform estimates allow passage through the iterated Cauchy integrals in all variables. The resulting multivariable Cauchy formula proves joint holomorphy. This establishes the first assertion.

Conjugating the homogeneous operator by $x^{\lambda }$ replaces $\theta$ by $\theta +\lambda$. After expansion it remains monic of order $N$, and its lower coefficients are finite constant linear combinations of the original $f_k$ and the constant function $1$. They retain continuity and disk holomorphy. It therefore suffices to prove the logarithmic assertion for $\lambda =0$. Write $y_j={\sum }_n{c}_{n,j}(b)x^n$ and $f_N=1$. The coefficient of $(\log x{)}^M$ in the equation is $P(\theta )y_M=0$, where $P(\theta )={\sum }_{k=0}^N{f}_k{\theta }^k$. The power-series result proves the absolute compact-uniform estimate, continuity, disk holomorphy and derivative properties for $y_M$. More generally, distributing the $k$ derivatives in ${\theta }^k(y_j(\log x{)}^j)$ shows that the coefficient of $(\log x{)}^i$ is $P(\theta )y_i=-{\sum }_{k=0}^N{f}_k{\sum }_{j=i+1}^{\min (M,i+k)}\left(\genfrac{}{}{0px}{}{k}{j-i}\right)\frac{j!}{i!}{\theta }^{k-j+i}{y}_j\mathrm{.}$ Exactly $j-i$ derivatives act on the logarithmic factor, giving the displayed factorial.

Descend from $i=M-1$ to $0$. The right-hand side in step 4.1 is a finite sum of continuous functions holomorphic in $x$ on the full disk: every $y_j$ appearing there has $j>i$, and its derivative property was established in the preceding descent step. It is therefore a valid inhomogeneity for the power-series result, which proves the same absolute estimate and regularity properties for $y_i$. Induction proves these conclusions for all $i$. On each compact subset of the cut punctured disk, $x^{\lambda }$ and the finitely many powers of its principal logarithm are bounded. Multiplying the finitely many coefficient estimates by these bounds proves absolute local uniform convergence of $Y$, and uniform convergence preserves continuity and the stated holomorphy. Analytic coefficient functions preserve each exponent class modulo integers. Thus a finite number of distinct classes satisfy their equations separately; applying this proof to each and summing their finite bounds proves the last assertion.