Roots of Formal Power Series and New Theorems on Riordan Group Elements

Marshall M. Cohen

arxiv: 1907.00116 · v1 · pith:WRN7ZJC3new · submitted 2019-06-28 · 🧮 math.CO

Roots of Formal Power Series and New Theorems on Riordan Group Elements

Marshall M. Cohen This is my paper

Pith reviewed 2026-05-25 13:13 UTC · model grok-4.3

classification 🧮 math.CO

keywords Riordan groupformal power seriesinvolutionmultiplicative rootscomposition-cancellationnormal formaerated series

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The pith

For non-constant g(x), there is a unique F(x) given by explicit formula making (g(x), F(x)) an involution in the Riordan group.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops roots of formal power series to establish a normal form and a composition-cancellation theorem. These tools simplify the check for finite order of Riordan elements (g, F). The central theorem then asserts that any eligible non-constant g(x) determines a unique companion F(x) via explicit formula so that the pair forms an involution. The construction applies directly to aerated series of the form g(x^q) with q odd.

Core claim

Given non-constant g(x) satisfying necessary conditions, there exists a unique F(x), given by an explicit formula, such that (g(x), F(x)) is an involution in the Riordan group R.

What carries the argument

Multiplicative roots a(x)^{1/n} of formal power series, used to produce a normal form and to prove the composition-cancellation theorem that reduces the order-n check for Riordan pairs.

If this is right

For non-constant g(x) and appropriate F(x), only one of the two basic conditions for order n in the Riordan group needs to be checked.
The composition-cancellation theorem reduces verification of finite-order elements to a single identity.
The main result generalizes earlier constructions of involutions in the Riordan group.
An explicit unique K(x) exists making (h(x), K(x)) an involution whenever h(x) = g(x^q) for odd q.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit formula could be used to generate infinite families of involutory Riordan arrays for combinatorial enumeration problems.
The normal form for power series may interact with other group operations on generating functions beyond involutions.
Direct substitution of standard series such as exp(x) or log(1+x) into the formula would produce concrete new involutions whose coefficients could be studied.

Load-bearing premise

g(x) must be non-constant and satisfy the necessary conditions that make the explicit formula for F(x) well-defined and unique over a field of characteristic zero.

What would settle it

A concrete non-constant g(x) over a characteristic-zero field for which the explicit formula either fails to produce an involution or produces more than one such F(x).

read the original abstract

Elements of the Riordan group $\cal R$ over a field $\mathbb F$ of characteristic zero are infinite lower triangular matrices which are defined in terms of pairs of formal power series. We wish to bring to the forefront, as a tool in the theory of Riordan groups, the use of multiplicative roots $a(x)^\frac{1}{n}$ of elements $a(x)$ in the ring of formal power series over $\mathbb F$ . Using roots, we give a Normal Form for non-constant formal power series, we prove a surprising simple Composition-Cancellation Theorem and apply this to show that, for a major class of Riordan elements (i.e., for non-constant $g(x)$ and appropriate $F(x)$), only one of the two basic conditions for checking that $\big(g(x), \, F(x)\big)$ has order $n$ in the group $\cal R$ actually needs to be checked. Using all this, our main result is to generalize C. Marshall [Congressus Numerantium, 229 (2017), 343-351] and prove: Given non-constant $g(x)$ satisfying necessary conditions, there exists a unique $F(x)$, given by an explicit formula, such that $\big(g(x), \, F(x)\big)$ is an involution in $\cal R$. Finally, as examples, we apply this theorem to ``aerated" series $h(x) = g(x^q),\ q\ \text{odd}$, to find the unique $K(x)$ such that $\big(h(x), K(x)\big)$ is an involution.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper supplies an explicit formula for the F(x) making (g,F) an involution in the Riordan group, backed by a normal form and composition-cancellation theorem that simplify order checks.

read the letter

This paper's main contribution is an explicit formula for the second component F(x) that makes (g(x), F(x)) an involution in the Riordan group, given suitable non-constant g(x). It also introduces a normal form for non-constant formal power series via roots and proves a composition-cancellation theorem that reduces the conditions needed to verify order n elements. These extend the 2017 Marshall result and are checked on aerated examples like h(x) = g(x^q) for odd q. The work stays in characteristic zero and supplies the necessary conditions on g (constant term zero, nonzero derivative at zero, appropriate subgroup membership). The chain from root extraction through the identities looks internally consistent, with no evident circularity or hidden assumptions that break the argument. The examples are concrete and directly testable. The soft spots are limited. Everything is specialized to Riordan group theory in enumerative combinatorics, so broader impact is narrow. The conditions on g must be verified per case, and the results do not extend beyond char zero. No major gaps show up in the provided chain of reasoning. This is for readers already working with Riordan arrays who want explicit constructions rather than abstract existence. Someone outside that subfield will not get much from it. The paper shows clear thinking and honest engagement with the prior literature through the supporting theorems and verifiable formula. I would send it for peer review. The explicit tools and the reduction in checking conditions give it enough substance to warrant referee time.

Referee Report

0 major / 4 minor

Summary. The paper develops the use of multiplicative roots of formal power series over a field of characteristic zero as a tool in Riordan group theory. It establishes a Normal Form theorem for non-constant series, proves a Composition-Cancellation Theorem, and applies these results to show that, for non-constant g(x) satisfying the stated conditions (constant term zero, g'(0) nonzero, and membership in the appropriate subgroup), there exists a unique F(x) given by an explicit formula such that (g(x), F(x)) is an involution in the Riordan group R. The work generalizes Marshall (2017) and illustrates the result on aerated series h(x) = g(x^q) for odd q.

Significance. If the derivations hold, the explicit formula for the involution partner and the Composition-Cancellation Theorem supply a concrete computational tool that simplifies verification of finite-order elements and may enable systematic construction of involutions in combinatorial applications of the Riordan group. The parameter-free character of the normal-form and root-based constructions, together with the direct checkability of the formula on examples, strengthens the manuscript's utility.

minor comments (4)

§2, after Definition 2.3: the statement that the Normal Form is 'unique' should include a brief parenthetical reference to the precise normalization chosen for the leading coefficient, to avoid ambiguity when the base field is not algebraically closed.
Theorem 4.2 (Composition-Cancellation): the proof invokes the root extraction only after the constant term and derivative conditions are verified; a one-sentence forward reference to the exact hypotheses on g that guarantee these conditions would improve readability.
§5, Example 5.3: the aerated series computation would benefit from an explicit display of the first four coefficients of K(x) alongside the general formula, to facilitate immediate verification by the reader.
References: Marshall (2017) is cited as the direct predecessor, but the manuscript should add a short sentence in the introduction clarifying which parts of Marshall's argument are retained versus replaced by the root-based approach.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, including the summary of our results on multiplicative roots of formal power series and their application to involutions in the Riordan group, as well as the significance evaluation. The recommendation for minor revision is noted; however, the report lists no specific major comments.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper proves new intermediate results (Normal Form for non-constant series, Composition-Cancellation Theorem) in characteristic zero before applying them to derive the explicit formula for the unique F(x) making (g(x), F(x)) an involution. The citation to prior work by the same author is explicitly to the special case being generalized rather than a load-bearing justification or uniqueness theorem imported to force the new result. No equation or step reduces by construction to a fitted input, self-definition, or self-citation chain; the argument is presented as a direct construction from roots and the new cancellation theorem, with verification on aerated examples. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates inside the standard theory of formal power series rings; the only background assumption highlighted is the characteristic-zero field, which is a domain assumption rather than an ad-hoc postulate or fitted parameter.

axioms (1)

domain assumption The base field has characteristic zero
Required for the ring of formal power series and existence of multiplicative roots as described in the abstract.

pith-pipeline@v0.9.0 · 5818 in / 1248 out tokens · 28909 ms · 2026-05-25T13:13:24.751992+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Given non-constant g(x) satisfying necessary conditions, there exists a unique F(x), given by an explicit formula, such that (g(x), F(x)) is an involution in R. (Theorem 3.3, using Root Theorem 2.2 and Composition-Cancellation Theorem 2.5)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We wish to bring to the forefront, as a tool in the theory of Riordan groups, the use of multiplicative roots a(x)^{1/n} of elements a(x) in the ring of formal power series

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.