REVIEW 2 major objections 5 minor 7 references
Square roots in the Appell group and Sprugnoli arrays
T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read A map from power-series pairs to Sprugnoli arrays yields square roots of aerated Appell arrays and new involutions in the Sprugnoli group.
desk verdict The intended square-root theorem is sound and the examples check out, but Proposition 13 as printed is missing a factor (1-x^2) in the numerator, so the paper needs a one-line correction before the central formula can be trusted as stated. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the mapping M(g,f), defined by the bivariate generating function B(x,y) = g(x)/((1-x)(1+yx)) + x f(x)/((1-$x^{2}$)(1-yx)). It sends pairs of power series to Sprugnoli arrays, and the key identity for its square reduces the squared generating function to a single Appell form. The Sprugnoli group product and the fundamental theorem for Sprugnoli arrays are the tools that carry the calculation.
What would settle it
Symbolically expand the two identities (G,F1,x)·G = (g(x)g(-x)(1-$x^{2}$)+$x^{2}$ f(x)^2)/(1-$x^{2}$)^2 and (G,F1,x)·GF1 = x(G,F1,x)·G for a generic even f(x) and generic g(x); any counterexample from the expansions would invalidate Proposition 13. Alternatively, compute a finite truncation of M(g,f)^2 for a concrete (g,f) with even f by direct matrix multiplication and compare to the Appell formula; a mismatch at any entry settles it.
Extended reading notes
Core claim
The central discovery is Proposition 13: when f(x) is an even power series, the square of the Sprugnoli array M(g,f) equals ((g(x)g(-x)+$x^{2}$ f(x)^2))/(1-$x^{2}$)^2, which is an aerated element of the Appell subgroup of the Riordan group. Consequently, every aerated Appell array that can be written in this form has a square root that is a Sprugnoli array M(g,f). If the right-hand side is the constant 1, then M(g,f) is an involution in the Sprugnoli group. The proof uses the Sprugnoli group product rule and the fundamental theorem of Sprugnoli arrays to compute the square componentwise.
Load-bearing premise
The proof of Proposition 13 assumes two algebraic simplifications for the components of M(g,f)^2 that are stated without expansion; if either identity fails, the square-root formula and the subsequent involution constructions collapse. It also assumes the Sprugnoli group product rules of [1] without re-derivation.
Editorial extensions
If this is right
- Every aerated Appell array expressible as (g(x)g(-x)+x^2 f(x)^2)/(1-x^2)^2 with f even has an explicit Sprugnoli square root M(g,f).
- If that expression equals 1, M(g,f) is an involution in the Sprugnoli group, generating new examples such as ((1+rx)/(1-x), -x(1+x+(r-1)x^2)/(1+rx), x)^2 = I.
- The mapping can be viewed as a map from the Riordan group to the Sprugnoli group via (g, xf) -> ((g/(1-x)+xf/(1-x^2), ...), and when xf is odd the image squares into the aerated Appell subgroup.
- This gives an explicit tool for constructing square roots of aerated Appell arrays, complementing earlier constructions of square roots for Bell matrices.
Reading between the lines
- The method likely extends beyond aerated Appell arrays: other Riordan arrays might have Sprugnoli square roots, a possibility the paper leaves open.
- The algebraic identities asserted 'by simplification' in Proposition 13 could be verified by direct symbolic expansion; if verified, the construction is fully explicit and parameterizes a family of involutions.
- The map (g, xf) -> M(g,f) may have additional structural properties, such as compatibility with products or preservation of combinatorial statistics, that could deepen the connection between the Riordan and Sprugnoli groups.
- A computational test on other simple aerated Appell arrays (e.g., 1/(1+x^2) or 1/(1-x^2-x^4)) would reveal which admit rational or polynomial choices for g and f, suggesting a classification of representable square roots.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces a mapping from pairs (g,f) of power series to the bivariate generating function B(x,y)=g(x)/((1-x)(1+yx))+xf(x)/((1-x^2)(1-yx)), whose coefficient matrix M(g,f) is shown to be a Sprugnoli array. The central result (Proposition 13) is that when f is an even power series, M(g,f)^2 is an aerated Appell array; this gives a constructive procedure for square roots of such arrays and, when the square equals the identity, yields involutions in the Sprugnoli group. The paper includes many worked examples and a Riordan-group interpretation of the construction.
Significance. If the corrected formula is used, the paper gives a clean, explicit construction of square roots for a class of aerated Appell arrays, with the square roots landing in the less familiar Sprugnoli group. This is a useful contribution to the Riordan-group toolkit. The main claims are concrete and falsifiable, and the worked examples provide genuine verification. The paper also correctly identifies a source of involutions in the Sprugnoli group. Its main limitation is that the Sprugnoli group machinery is imported from the author's previous preprint [1], and the proof of the key proposition contains algebraic simplifications that are only asserted.
major comments (2)
- [§6, Proposition 13] The displayed square formula is missing a factor (1-x^2) in the numerator. As printed it reads M(g,f)^2 = (g(x)g(-x)+x^2 f(x)^2)/(1-x^2)^2, but the proof's own simplification gives (g(x)g(-x)(1-x^2)+x^2 f(x)^2)/(1-x^2)^2. The printed statement is false: for g=1, f=0, M(1,0)=(1/(1-x), -x) and its square is (1/(1-x^2), x), not (1/(1-x^2)^2, x). Every later use (Corollary 14, Examples 15–18, Section 7) uses the corrected numerator. Please restore the factor throughout the displayed Proposition.
- [§6, proof of Proposition 13] The proof of Proposition 13 relies on two nontrivial algebraic identities, introduced by 'Simplifying this, we find' and 'With these expressions, we find': the formula for (G,F1,x)·G and the identity (G,F1,x)·GF1 = x(G,F1,x)·G. These are load-bearing for the central claim. Please include the intermediate expansions or a concise derivation for both, so the proof can be checked without redoing the computation. An independent expansion shows the identities are correct, so this is a presentation/verification issue rather than a fundamental error.
minor comments (5)
- [§2, Example 2] The matrix displays in this example are difficult to read because of missing spacing/line breaks, especially the right-hand matrix. Please typeset them more carefully.
- [§4] The product and inverse of Sprugnoli arrays are quoted from [1] without proof. Since [1] is an arXiv preprint, a brief statement of the group law or a citation to a published version would improve self-containedness.
- [§6] The notation G e(x^2), G o(x^2) is ambiguous. It should be written as (G_e)(x^2) and (G_o)(x^2), with a short clarification that these are the even/odd bisections of G.
- [§8] The sentence 'if the element (g, xf)∈R is such that xf is odd' is slightly confusing: since x is odd, xf is odd exactly when f is even. Please phrase it to match the earlier hypothesis directly.
- [References] Reference [1] is to an arXiv preprint. If a journal version or a more stable published source exists, it would be preferable to cite that.
Circularity Check
No significant circularity: the square-root formula is a direct computation from the defining generating function; the only self-citation is background group structure.
full rationale
The derivation is self-contained in the relevant sense. Proposition 8 verifies that the bivariate generating function B(x,y) defining M(g,f) is the generating function of a Sprugnoli array by algebraically rewriting it in the canonical form g(x)(1+y f1(x))/(1-y^2 x f2(x)); this is a direct computation, not a definitional shortcut. Proposition 13 then uses the Sprugnoli product rule, stated in Section 4, to compute M(g,f)^2. That product rule is prior group-theoretic background from the author's earlier paper [1], but it is parameter-free, does not assume the square-root result, and is stated explicitly in the text, so its use is not a circular reduction of the target theorem. The evenness of f is a genuine hypothesis, not a fitted parameter, and the resulting expression (g(x)g(-x)(1-x^2)+x^2f(x)^2)/(1-x^2)^2 is derived from the product rule. The proof does contain two unexpanded algebraic identities introduced by 'Simplifying this, we find' and 'With these expressions, we find,' but these are computational gaps, not instances of assuming the conclusion; they are independently checkable and do not make the theorem equivalent to its inputs. The later examples solve explicit equations for g(x) given a desired aerated Appell array, so no fitted value is later renamed as a prediction. One non-circular flaw is that the display in Proposition 13 omits the factor (1-x^2) in the numerator that both the proof and every worked example use; this is a typographical/correctness issue, not a circularity. Overall, no step in the claimed derivation reduces by construction to its own inputs.
Assumptions & free parameters
assumptions (4)
- standard math Riordan group product and inverse formulas: (g,f).(u,v) = (g u(f), v(f)), inverse uses compositional inverse.
- domain assumption Sprugnoli group product, inverse, and fundamental theorem of Sprugnoli arrays as stated in [1].
- standard math Formal power series over C with F0 multiplicatively invertible and F1 compositionally invertible.
- standard math When f is even, f(-x)=f(x).
Cite this review
Pith. "Pith review of Square roots in the Appell group and Sprugnoli arrays." pith.science (2026). https://pith.science/paper/645F455S
@misc{pith2026260801497,
author = {Pith},
title = {Pith review of: Square roots in the Appell group and Sprugnoli arrays},
year = {2026},
howpublished = {\url{https://pith.science/paper/645F455S}},
note = {Machine review of arXiv:2608.01497}
}
read the original abstract
We introduce a special mapping from pairs of power series to the group of Sprugnoli matrices. This mapping has the property when the second argument is an even power series, then the square of the resulting Sprugnoli array is an aerated element of the Appell subgroup of the Riordan group. This allows us to explore the square roots of elements in the aerated Appell subgroup. As the identity is an element of this subgroup, we are led to explore related involutions in the Sprugnoli group.
Reference graph
Works this paper leans on
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[1]
P. Barry, A new group in the Riordan family of matrix groups: the Sprugnoli group, https://arxiv.org/abs/2605.16633, 2026
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[2]
Barry,Riordan Arrays: a Primer, Logic Press, 2017
P. Barry,Riordan Arrays: a Primer, Logic Press, 2017
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Merlini, On the square root of a Bell matrix,Results in Mathematics,76(2021), Article 46, 18 pages
D. Merlini, On the square root of a Bell matrix,Results in Mathematics,76(2021), Article 46, 18 pages
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[4]
L. Shapiro, R. Sprugnoli, P. Barry, G.-S. Cheon, T.-X. He, D. Merlini, and W. Wang, The Riordan Group and Applications, Springer, 2022. 15
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L. W. Shapiro, S. Getu, W. J. Woan, and L. C. Woodson, The Riordan group,Discr. Appl. Math.34(1991), 229–239
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N. J. A. Sloane,The On-Line Encyclopedia of Integer Sequences. Published electronically athttp://oeis.org, 2026
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[7]
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences,Notices Amer. Math. Soc.50(2003), 912–915. 2020Mathematics Subject Classification: Primary 15B36; Secondary 05A15, 11B83, 11C20, 15A15. Keywords:Riordan array, Appell subgroup, Sprugnoli group, generating function, square root, involution. (Concerned with sequences A000045 , A052952, A074331, ...
work page 2003
Reviewed August 6, 2026 · model on record in the stance chip above.
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