D.3 Unit triangular coefficient transport
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The examples later in the paper use different kernels, but their coefficient algebra has one common form. This section records that form once. Let \(R\) be a commutative algebra over \({\mathbb Q}\), and take all coefficients in \(R\).
Let
There is a unique inverse
Historical theorem D.16: Exact coefficient transport
Let
Then
The inverse coordinate formula is
Both transforms are lower triangular with diagonal one.
Proof
The first identity is the coefficient rule for the product of exponential generating functions. Multiply by \(K(t)\) and use \(K(t)H(t)=1\) for the inverse identity. The diagonal coefficient in each transform is the constant coefficient of its kernel, which is one.
Historical corollary D.17: Composition of coefficient transports
If \(U_H\) and \(U_G\) are the triangular transforms associated with unit kernels \(H\) and \(G\), then
In particular, the inverse of \(U_H\) is \(U_{H^{-1}}\).
Proof
Multiplication by \(G\), followed by multiplication by \(H\), is multiplication by \(HG\).
Historical theorem D.18: Filtered triangular reconstruction
Let \(p\) be prime. Suppose
where \(c_r\in{\mathbb Z_p}\), \(S_n\in{\mathbb Z_p}\), and
For every precision \(L\geq1\),
When the omitted set is nonempty, the tail has valuation at least
Proof
Every omitted summand belongs to \(p^L{\mathbb Z_p}\). The sharper statement follows by taking the minimum of the exact lower bounds for the summands and applying the nonarchimedean triangle inequality.
Historical remark D.19
An arbitrary invertible triangular matrix can encode an arbitrary sequence. The transforms retained here have an explicit generating kernel and a visible filtration. Those two features make exact inversion and uniform truncation possible.