Concise and elegant proofs of three formulas for complete Bell polynomials

Feng Qi

arxiv: 2605.22959 · v1 · pith:FE5SU567new · submitted 2026-05-21 · 🧮 math.CO · math.NT

Concise and elegant proofs of three formulas for complete Bell polynomials

Feng Qi This is my paper

Pith reviewed 2026-05-25 05:36 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords complete Bell polynomialsgenerating functionscombinatorial identitiesexponential generating functionspolynomial formulasset partitionscombinatorial proofs

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The pith

The generating function of complete Bell polynomials directly yields concise proofs for three of their formulas.

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

The paper sets out to derive three formulas for complete Bell polynomials from their generating function in a short and direct manner. A reader would care because these polynomials encode the structure of set partitions and appear in expansions throughout combinatorics and analysis. The proofs replace longer inductions or explicit summations with manipulations that start from the generating function. This makes the identities easier to verify and remember.

Core claim

In light of the generating function of the complete Bell polynomials and other techniques, concise and elegant proofs are presented for three formulas for the complete Bell polynomials.

What carries the argument

The exponential generating function for the complete Bell polynomials, used to obtain the formulas by coefficient extraction or differentiation.

Load-bearing premise

The generating function together with the other techniques is enough to reach the three formulas without gaps or extra assumptions.

What would settle it

A direct numerical check that shows one of the three formulas fails for a small explicit choice of the variables or index.

read the original abstract

In the paper, in light of the generating function of the complete Bell polynomials and other techniques, the author presents concise and elegant proofs of three formulas for the complete Bell polynomials.

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

The paper supplies three new proofs of known formulas for complete Bell polynomials via the standard exponential generating function.

read the letter

The paper's main contribution is three concise proofs of existing formulas for complete Bell polynomials, derived from the standard exponential generating function. The author uses the generating function and other standard techniques to derive the formulas. It does a good job keeping the arguments short by relying on basic algebraic steps rather than heavy machinery. That matches the title's claim of elegance. If the full derivations are as brief as promised, they could serve as useful references. The limitation is clear: no new formulas or results, just alternative proofs. The topic is narrow, so impact stays within a small subfield of combinatorics. The abstract gives no sample steps, which makes it tough to assess how much simpler these proofs really are compared to prior work. Since the formulas are classical, the value hinges entirely on whether these versions are noticeably cleaner. This kind of paper suits readers who need clean derivations for these specific identities, perhaps for teaching or quick reference. It won't change how most people approach Bell polynomials or find new applications. I would take it to a reading group only if the group has a combinatorics or special functions focus. I wouldn't cite it myself in the next year. Still, the approach looks internally consistent with no obvious gaps or circularity, so it should go to peer review for verification of the derivations and to confirm they are indeed concise.

Referee Report

0 major / 2 minor

Summary. The manuscript presents concise and elegant proofs of three formulas for the complete Bell polynomials, relying on the exponential generating function for these polynomials together with standard algebraic techniques and manipulations.

Significance. Complete Bell polynomials appear frequently in combinatorics, algebra, and probability; streamlined proofs of their explicit formulas can improve accessibility and facilitate applications. The approach uses the canonical generating function, which is a standard and non-circular tool, and the claim of conciseness is a modest but potentially useful contribution if the derivations are gap-free.

minor comments (2)

The abstract does not name the three formulas; adding their explicit statements (or equation numbers from the literature) would help readers immediately identify the results being reproved.
Notation for the complete Bell polynomials and the generating function should be introduced with a brief reminder of the standard definition in §1 to make the paper self-contained for a broader audience.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; standard generating function yields independent proofs

full rationale

The paper states that it derives three formulas for complete Bell polynomials from the standard exponential generating function together with algebraic manipulations and other techniques. No equations reduce a claimed prediction to a fitted input by construction, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled via prior work by the same author. The central derivations remain self-contained once the generating function is accepted as given; the abstract and description give no indication that any load-bearing step collapses to a renaming or self-referential definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard generating function for complete Bell polynomials as background; no free parameters, new axioms, or invented entities are mentioned in the abstract.

axioms (1)

domain assumption The generating function of the complete Bell polynomials holds as previously established in the literature.
Invoked directly in the abstract as the foundation for the proofs.

pith-pipeline@v0.9.0 · 5529 in / 949 out tokens · 29423 ms · 2026-05-25T05:36:23.769897+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

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work page doi:10.1142/9634 2016

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Abramowitz and I

M. Abramowitz and I. A. Stegun (Eds),Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series55, Reprint of the 1972 edition, Dover Publications, Inc., New York, 1992

work page 1972

[2] [2]

Bao, C.-Y

L.-Y. Bao, C.-Y. He, and F. Qi,Monotonic sequences and inequalities involving the ratio between two adjacent nonzero Bernoulli numbers, Math. Inequal. Appl.29(2026), no. 1, 1–14. DOI:https://doi.org/10.7153/mia-2026-29-01

work page doi:10.7153/mia-2026-29-01 2026

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I. Chit ¸escu,Around the formula of Fa` a di Bruno,´Editions universitaires europ´ eennes, Mau- ritius, 2017

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Comtet,Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D

L. Comtet,Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974. DOI:https: //doi.org/10.1007/978-94-010-2196-8

work page doi:10.1007/978-94-010-2196-8 1974

[6] [6]

Genˇ cev,Extension of Hoffman’s combinatorial identity via specific zeta-like series, Re- sults Math.79(2024), no

M. Genˇ cev,Extension of Hoffman’s combinatorial identity via specific zeta-like series, Re- sults Math.79(2024), no. 1, Paper No. 2, 25 pages. DOI:https://doi.org/10.1007/ s00025-023-02035-w

work page 2024

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J. I. B. Gil and J. Fres´ an,Multiple Zeta Values: from Numbers to Motives, Clay Mathematics Proceedings, 2020. URL:https://javier.fresan.perso.math.cnrs.fr/mzv.pdf

work page 2020

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I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Elsevier/Academic Press, Amsterdam, 2015. DOI:https://doi.org/10.1016/B978-0-12-384933-5.00013-8

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He and F

C.-Y. He and F. Qi,Reformulations and generalizations of Hoffman’s and Genˇ cev’s com- binatorial identities, Results Math.79(2024), no. 4, Paper No. 131, 17 pages. DOI: https://doi.org/10.1007/s00025-024-02160-0

work page doi:10.1007/s00025-024-02160-0 2024

[11] [11]

M. E. Hoffman,Multiple harmonic series, Pacific J. Math.152(1992), no. 2, 275–290. URL: http://projecteuclid.org/euclid.pjm/1102636166

work page arXiv 1992

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Koshy,Catalan Numbers with Applications, Oxford University Press, Oxford, 2009

T. Koshy,Catalan Numbers with Applications, Oxford University Press, Oxford, 2009

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Sprugnoli,Riordan Array Proofs of Identities in Gould’s Book, University of Florence, Italy, 2006

R. Sprugnoli,Riordan Array Proofs of Identities in Gould’s Book, University of Florence, Italy, 2006

work page 2006

[14] [14]

F. Qi,Alternative proofs of Xu’s forms for He–Qi’s combinatorial identities reformulating and generalizing Hoffman’s and Genˇ cev’s combinatorial identities for complete Bell polyno- mials, Filomat40(2026), accepted on 10 February 2026. DOI:https://www.researchgate. net/publication/400661711

work page arXiv 2026

[15] [15]

Qi,Some properties of the Catalan numbers, Appl

F. Qi,Some properties of the Catalan numbers, Appl. Anal. Discrete Math.19(2025), no. 1, 176–184. DOI:https://doi.org/10.2298/AADM240130002Q

work page doi:10.2298/aadm240130002q 2025

[16] [16]

Qi and B.-N

F. Qi and B.-N. Guo,Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterr. J. Math.14(2017), no. 3, Article 140, 14 pages. DOI:https://doi.org/10.1007/s00009-017-0939-1

work page doi:10.1007/s00009-017-0939-1 2017

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F. Qi, G. V. Milovanovi´ c, and D. Lim,Specific values of partial Bell polynomials and series expansions for real powers of functions and for composite functions, Filomat37(2023), no. 28, 9469–9485. DOI:https://doi.org/10.2298/FIL2328469Q. 16 F. QI

work page doi:10.2298/fil2328469q 2023

[18] [18]

Qi, D.-W

F. Qi, D.-W. Niu, D. Lim, and B.-N. Guo,Closed formulas and identities for the Bell poly- nomials and falling factorials, Contrib. Discrete Math.15(2020), no. 1, 163–174. DOI: https://doi.org/10.11575/cdm.v15i1.68111

work page doi:10.11575/cdm.v15i1.68111 2020

[19] [19]

Qi and P

F. Qi and P. Taylor,Series expansions for powers of sinc function and closed-form expres- sions for specific partial Bell polynomials, Appl. Anal. Discrete Math.18(2024), no. 1, 92–115. DOI:https://doi.org/10.2298/AADM230902020Q

work page doi:10.2298/aadm230902020q 2024

[20] [20]

Quaintance and H

J. Quaintance and H. W. Gould,Combinatorial Identities for Stirling Numbers, the un- published notes of H. W. Gould, with a foreword by George E. Andrews, World Scientific Publishing Co. Pte. Ltd., Singapore, 2016

work page 2016

[21] [21]

Riordan,Combinatorial Identities, Reprint of the 1968 original, Robert E

J. Riordan,Combinatorial Identities, Reprint of the 1968 original, Robert E. Krieger Pub- lishing Co., Huntington, N.Y., 1979

work page 1968

[22] [22]

R. P. Stanley,Catalan Numbers, Cambridge University Press, New York, 2015. DOI:https: //doi.org/10.1017/CBO9781139871495

work page doi:10.1017/cbo9781139871495 2015

[23] [23]

N. M. Temme,Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996. DOI: http://dx.doi.org/10.1002/9781118032572

work page doi:10.1002/9781118032572 1996

[24] [24]

Xu,Concise forms of He–Qi combinatorial identities in terms of generalized Bernoulli polynomials, Euler polynomials, and Catalan numbers, ResearchGate Preprint (2026)

A. Xu,Concise forms of He–Qi combinatorial identities in terms of generalized Bernoulli polynomials, Euler polynomials, and Catalan numbers, ResearchGate Preprint (2026). DOI: https://doi.org/10.13140/RG.2.2.14137.63849

work page doi:10.13140/rg.2.2.14137.63849 2026

[25] [25]

Zagier,Values of zeta functions and their applications, First European Congress of Math- ematics, Vol

D. Zagier,Values of zeta functions and their applications, First European Congress of Math- ematics, Vol. II (Paris, 1992), 497–512, Progr. Math., 120, Birkh¨ auser, Basel, 1994

work page 1992

[26] [26]

Zhao,Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, Se- ries on Number Theory and its Applications, 12, World Scientific Publishing Co

J. Zhao,Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, Se- ries on Number Theory and its Applications, 12, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. DOI:https://doi.org/10.1142/9634. 17709 Sabal Court, University Village, Dallas, TX 75252-8024, USA Email address:qifeng618@gmail.com URL:https://orcid.org/0...

work page doi:10.1142/9634 2016