The truncated binomial family $T_h^{s,N}$, the truncated logarithm $L_N(h)$, its coefficient fields $X_j$ and the truncations $X^{[N]}$

Let $d\ge 1$, let $(U,U_{\rho },\tilde{A},\tilde{B},M,K,\delta ,{\mathrm{\Psi }}_h)$ be analytic splitting data (Analytic splitting data with uniform holomorphic neighbourhood, and the exact symmetric splitting map ${\mathrm{\Psi }}_h$), let $C_0=C_0(d)$ be the constant of The difference operator ${\mathrm{\Delta }}_h={\mathrm{\Psi }}_h^{\ast }-\mathrm{i}\mathrm{d}$ loses a controlled domain margin and is bounded by $C_{d}M\mathrm{\mid }h\mathrm{\mid }\mathrm{/}b$, and fix the notation $D(r)$, $b=\rho \mathrm{/}(8N)$, $r_N=\rho \mathrm{/}(C_{0}MN)$ and $D_k=D(\rho \mathrm{/}2-2bk)$ for $0\le k\le N$ of that lemma. Throughout, $N\ge 2$ is an integer, $\mathrm{\mid }h\mathrm{\mid }\le r_N$, and all functions are read on the domain $D_N$ unless stated otherwise.

Binomial coefficients. For $s\in \mathbb{C}$ and $k\in \mathbb{N}$ define

$\left(\genfrac{}{}{0px}{}{s}{k}\right):=\frac{{\prod }_{i=0}^{k-1}(s-i)}{k!},$

the empty product being $1$, and $k!$ the factorial of The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$ embedded in $\mathbb{C}$; thus $\left(\genfrac{}{}{0px}{}{s}{0}\right)=1$, $\left(\genfrac{}{}{0px}{}{s}{k}\right)$ is a polynomial in $s$ of degree $k$ with rational coefficients, $\left(\genfrac{}{}{0px}{}{m}{k}\right)$ agrees with the number of $k$-element subsets of an $m$-element set for integers $m\ge 0$ (and equals $0$ when $k>m$), and

${\frac{d}{ds}\left(\genfrac{}{}{0px}{}{s}{k}\right)\mid }_{s=0}=\frac{(-1{)}^{k+1}}{k}(k\ge 1),{\frac{d}{ds}\left(\genfrac{}{}{0px}{}{s}{0}\right)\mid }_{s=0}=0.$

Difference operator and pullbacks. For $f$ holomorphic on $D_0$ put ${\mathrm{\Delta }}_{h}f:=f\circ {\mathrm{\Psi }}_h-f$, a holomorphic function on $D_1$ by The difference operator ${\mathrm{\Delta }}_h={\mathrm{\Psi }}_h^{\ast }-\mathrm{i}\mathrm{d}$ loses a controlled domain margin and is bounded by $C_{d}M\mathrm{\mid }h\mathrm{\mid }\mathrm{/}b$; for $0\le k\le N$ let ${\mathrm{\Delta }}_h^{k}f$ be the $k$-fold iterate, holomorphic on $D_k$, with ${\mathrm{\Delta }}_h^{0}f:=f$. For $m\in \mathbb{N}$ let ${\mathrm{\Psi }}_h^m$ be the partial $m$-fold composite of ${\mathrm{\Psi }}_h$, defined only where every intermediate composition exists (so ${\mathrm{\Psi }}_h^0=\mathrm{i}\mathrm{d}$).

Truncated binomial family. For $s\in \mathbb{C}$, $f$ holomorphic on $D_0$ and $Z\in D_N$ define

$T_h^{s,N}f:={\sum }_{k=0}^N\left(\genfrac{}{}{0px}{}{s}{k}\right){\mathrm{\Delta }}_h^{k}f,$

a finite sum of functions each holomorphic on $D_k\supseteq D_N$: it is a holomorphic function on $D_N$, and for fixed $h$ and $f$ the map $s\mapsto T_h^{s,N}f$ is a polynomial of degree at most $N$ with values in the holomorphic functions on $D_N$.

Truncated logarithm and coefficient fields. The truncated logarithm is

$L_N(h)f:={\sum }_{k=1}^N\frac{(-1{)}^{k+1}}{k}{\mathrm{\Delta }}_h^{k}f=T_h^{s,N}f{\mid }_{s=0}\text{ differentiated in }s,$

that is, $L_N(h)f=\frac{d}{ds}\left(T_h^{s,N}f\right){\mid }_{s=0}$; it is holomorphic on $D_N$. For $1\le j\le N$ and each coordinate function $z_i$ ($i=1,\dots ,2d$), the function $h\mapsto (L_N(h)z_i)(Z)$ is holomorphic on $\{\mathrm{\mid }h\mathrm{\mid }\le r_N\}$ and vanishes at $h=0$ (each ${\mathrm{\Delta }}_h^k{z}_i$ is $O(h^k)$), so the quotient $(L_N(h)z_i)(Z)\mathrm{/}h$ has a removable singularity at $h=0$ and a well-defined Taylor coefficient of $h^{j-1}$; define the $j$-th coefficient field $X_j:D_N\to {\mathbb{C}}^{2d}$ by

$(X_j{)}_i(Z):=[h^{j-1}]\frac{(L_N(h)z_i)(Z)}{h}(Z\in D_N,i=1,\dots ,2d),$

the coefficient being extracted by Cauchy's formula in the one variable $h$ (Calculus of holomorphic maps: chain rule, Wirtinger derivatives, coefficient extraction in one variable, and locally uniform limits clause 3). Finally the truncation of the modified field at order $N$ is

$X^{[N]}(h):={\sum }_{j=1}^N{h}^{j-1}{X}_j,$

a holomorphic map $D_N\to {\mathbb{C}}^{2d}$ for every $\mathrm{\mid }h\mathrm{\mid }\le r_N$.

• The definitions are coordinate-wise and choice-free; the coefficient extraction in $h$ is the only analytic operation, and it is justified by the one-variable holomorphy in $h$ supplied by The difference operator ${\mathrm{\Delta }}_h={\mathrm{\Psi }}_h^{\ast }-\mathrm{i}\mathrm{d}$ loses a controlled domain margin and is bounded by $C_{d}M\mathrm{\mid }h\mathrm{\mid }\mathrm{/}b$ and Hamiltonian flows and the exact splitting map preserve the complexified canonical form and the Poisson bracket.
• The following facts are not part of the definition and are proved on this page: for integers $0\le m\le N$ one has $T_h^{m,N}f=f\circ {\mathrm{\Psi }}_h^m$ on $D_m$ (The binomial family of the splitting pullback is multiplicative and bracket-preserving, and its logarithm is a derivation); the fields $X_j$ are independent of the truncation order in the sense that the coefficient of $h^{j-1}$ in $L_N(h)$ coincides with that in $L_{N^{\mathrm{\prime }}}(h)$ on $D(\rho \mathrm{/}4)$ for $N,N^{\mathrm{\prime }}\ge j$ (Bounds for the truncated logarithm: $\mid L_N(h)z_i\mathrm{/}h\mid \le 8M\mathrm{/}3$ and $\mid X_j\mid \le (8M\mathrm{/}3)(C_{0}Mj\mathrm{/}\rho {)}^{j-1}$); each $X_j$ is a holomorphic vector field and a symplectic one (Each coefficient field $X_j$ of the modified logarithm is a holomorphic symplectic field with a global holomorphic primitive $H_j$ on the star-shaped domain, and $X_1=X_{\tilde{A}+\tilde{B}}$); and $X_1=X_{\tilde{A}+\tilde{B}}$.