Combinatorial sums derived from properties of Legendre polynomials

Michel Bataille; Robert Frontczak

arxiv: 2604.25981 · v2 · submitted 2026-04-28 · 🧮 math.NT

Combinatorial sums derived from properties of Legendre polynomials

Michel Bataille , Robert Frontczak This is my paper

Pith reviewed 2026-05-07 14:42 UTC · model grok-4.3

classification 🧮 math.NT MSC 05A1933C45

keywords combinatorial sumsLegendre polynomialsclosed formsintegralsspecial functionsbinomial identities

0 comments

The pith

An identity with Legendre polynomials yields closed forms for multiple combinatorial sums.

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

The paper starts from an identity relating a combinatorial sum to Legendre polynomials. It uses this link, along with integral properties of associated functions, to obtain explicit closed-form expressions for a number of combinatorial sums. A sympathetic reader would care because such sums often arise in counting problems and number theory, where direct closed forms replace recursive or numerical evaluation. The work shows how special-function identities can simplify these expressions in concrete cases.

Core claim

From an identity connecting a combinatorial sum and Legendre polynomials, we derive closed forms for a number of combinatorial sums. Some of them are obtained via results about the integrals of functions associated with Legendre polynomials.

What carries the argument

The identity connecting a combinatorial sum to Legendre polynomials, together with integral evaluations of associated functions.

If this is right

Explicit closed forms are obtained for various combinatorial sums linked to the base identity.
Integral results for Legendre-associated functions supply additional closed forms.
The method extends the evaluation of sums by connecting them to properties of these polynomials.

Where Pith is reading between the lines

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

The closed forms may simplify proofs of related binomial identities or congruences.
Analogous identities with other orthogonal polynomials could produce closed forms for different classes of sums.
The approach offers a template for turning generating-function sums into direct evaluations.

Load-bearing premise

The initial identity equating the combinatorial sum to an expression involving Legendre polynomials holds, and the relevant integrals can be evaluated in closed form.

What would settle it

Compute a specific combinatorial sum numerically for small parameters and check whether it equals the proposed closed-form expression involving Legendre polynomials or their integrals.

read the original abstract

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 derives closed forms for several combinatorial sums by starting from a Legendre polynomial identity and using integral evaluations of associated functions.

read the letter

The core of this paper is straightforward: it begins with an identity tying a combinatorial sum to Legendre polynomials and then extracts closed forms for a collection of related sums, with some obtained through integrals of Legendre-associated functions. The specific sums and the resulting expressions appear to be the new part, as they are not presented as previously known in the abstract framing. The derivations stick to standard properties such as orthogonality and generating functions, which keeps the work grounded and reproducible in principle. This is a clean application rather than a reinvention of tools. The paper does well by delivering explicit formulas that could be checked or used directly in calculations involving binomial or hypergeometric sums. The integral route for some cases adds a useful layer without overcomplicating things. Soft spots are limited and proportionate. The starting identity is the load-bearing piece, so the paper needs to establish it clearly without gaps; if it does, the rest follows from routine manipulations. There is no sign of circular reasoning or unstated convergence problems in the central construction. The overall contribution stays incremental, which is fine but means it will mainly interest specialists rather than a wider audience. This is the sort of paper for readers in number theory or combinatorics who hunt for closed forms in sums that can be linked to orthogonal polynomials. Someone needing concrete expressions for particular cases might cite or apply it, but it is not likely to reorganize the field. It deserves a serious referee because the claims are specific, the methods are standard and verifiable, and the results are falsifiable through direct computation. I would send it to peer review rather than desk reject.

Referee Report

0 major / 3 minor

Summary. The manuscript starts from a stated identity linking a combinatorial sum to Legendre polynomials and derives closed-form expressions for a number of combinatorial sums, with some obtained by evaluating integrals of functions associated with Legendre polynomials.

Significance. If the starting identity is valid and the algebraic and integral manipulations are rigorous, the work establishes a direct bridge between combinatorial sums and standard properties of Legendre polynomials (orthogonality, Rodrigues formula, generating functions). This could furnish new closed forms in number theory and combinatorics; the analytic component via integrals is a notable strength.

minor comments (3)

[Abstract] The abstract is brief and general; naming the specific combinatorial sums treated (e.g., by their summation indices or generating functions) would immediately clarify the paper's scope and novelty.
When invoking standard properties of Legendre polynomials, include explicit citations to the relevant theorems or formulas (e.g., orthogonality integral or Rodrigues formula) at each derivation step to aid verification.
Check that all integral evaluations are accompanied by justification of convergence or contour choices if the associated functions are not entire.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary and significance statements, and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

Derivation is self-contained from stated identity and standard Legendre properties

full rationale

The paper opens with an explicit identity connecting a combinatorial sum to Legendre polynomials and proceeds via direct algebraic manipulations, generating functions, Rodrigues formula, and integral evaluations that rely on classical, externally established properties of Legendre polynomials (orthogonality, recurrence relations). No step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation whose validity depends on the present work. The central claims remain independent of the paper's own outputs, satisfying the default expectation of no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard properties of Legendre polynomials (orthogonality, generating functions, integral representations) and basic combinatorial identities; no free parameters, invented entities, or ad-hoc axioms introduced.

axioms (1)

standard math Legendre polynomials satisfy their standard differential equation, orthogonality relations, and integral identities.
Invoked implicitly to connect combinatorial sums and to evaluate integrals as described in the abstract.

pith-pipeline@v0.9.0 · 5304 in / 1053 out tokens · 53201 ms · 2026-05-07T14:42:43.099439+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Aloui, Legendre polynomials in terms of integrals involving Hermite polynomials, Period

B. Aloui, Legendre polynomials in terms of integrals involving Hermite polynomials, Period. Math. Hung. 91 (2025), 112–123

work page 2025
[2]

Bataille, Solution to 1996 Ukrainian Mathematical Olympiad Problem 8, Crux Math

M. Bataille, Solution to 1996 Ukrainian Mathematical Olympiad Problem 8, Crux Math. 27 (7) (2001), 428–429

work page 1996
[3]

Chu and J

W. Chu and J. M. Campbell, Expansions over Legendre polynomials and infinite double series identities, Ramanujan J. 60 (2023), 317–353

work page 2023
[4]

Diekemaa and T

E. Diekemaa and T. H. Koornwinder, Generalizations of an integral for Legendre polynomials by Persson and Strang, J. Math. Anal. Appl. 388 (2012), 125–135

work page 2012
[5]

Gautschi, On the preceding paper ”A Legendre Polynomial Integral” by James L

W. Gautschi, On the preceding paper ”A Legendre Polynomial Integral” by James L. Blue, Math. Comp. 146 (1979), 742–743

work page 1979
[6]

Gradshteyn and I

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, Elsevier Academic Press, 2007

work page 2007
[7]

V. J. W. Guo, Some congruences involving powers of Legendre polynomials, Integral Transforms Spec. Funct. 26 (2015), 660–666

work page 2015
[8]

H. W. Gould, Combinatorial Identities, Published by the author, Revised edition, 1972

work page 1972
[9]

A. D. Klemm and S. Y. Larsen, Some integrals involving Legendre polynomials providing combinatorial identities, J. Austral. Math. Soc. Ser. B 32 (1991), 304–310

work page 1991
[10]

N. J. A. Sloane,The On-Line Encyclopedia of Integer Sequences, https://oeis.org

work page
[11]

H. M. Srivastava and J. Choi,Series Associated with the Zeta and Related Functions, Springer Sci- ence+Media, B.V., 2001

work page 2001
[12]

Wan and W

J. Wan and W. Zudilin, Generating functions of Legendre polynomials: a tribute to Fred Brafman, J. Approx. Theory 164 (2012), 488–503. Independent Researcher, 76520 Franqueville-Saint-Pierre, France Email address:michelbataille@wanadoo.fr Independent Researcher, 72764 Reutlingen, Germany Email address:robert.frontczak@web.de

work page 2012

[1] [1]

Aloui, Legendre polynomials in terms of integrals involving Hermite polynomials, Period

B. Aloui, Legendre polynomials in terms of integrals involving Hermite polynomials, Period. Math. Hung. 91 (2025), 112–123

work page 2025

[2] [2]

Bataille, Solution to 1996 Ukrainian Mathematical Olympiad Problem 8, Crux Math

M. Bataille, Solution to 1996 Ukrainian Mathematical Olympiad Problem 8, Crux Math. 27 (7) (2001), 428–429

work page 1996

[3] [3]

Chu and J

W. Chu and J. M. Campbell, Expansions over Legendre polynomials and infinite double series identities, Ramanujan J. 60 (2023), 317–353

work page 2023

[4] [4]

Diekemaa and T

E. Diekemaa and T. H. Koornwinder, Generalizations of an integral for Legendre polynomials by Persson and Strang, J. Math. Anal. Appl. 388 (2012), 125–135

work page 2012

[5] [5]

Gautschi, On the preceding paper ”A Legendre Polynomial Integral” by James L

W. Gautschi, On the preceding paper ”A Legendre Polynomial Integral” by James L. Blue, Math. Comp. 146 (1979), 742–743

work page 1979

[6] [6]

Gradshteyn and I

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, Elsevier Academic Press, 2007

work page 2007

[7] [7]

V. J. W. Guo, Some congruences involving powers of Legendre polynomials, Integral Transforms Spec. Funct. 26 (2015), 660–666

work page 2015

[8] [8]

H. W. Gould, Combinatorial Identities, Published by the author, Revised edition, 1972

work page 1972

[9] [9]

A. D. Klemm and S. Y. Larsen, Some integrals involving Legendre polynomials providing combinatorial identities, J. Austral. Math. Soc. Ser. B 32 (1991), 304–310

work page 1991

[10] [10]

N. J. A. Sloane,The On-Line Encyclopedia of Integer Sequences, https://oeis.org

work page

[11] [11]

H. M. Srivastava and J. Choi,Series Associated with the Zeta and Related Functions, Springer Sci- ence+Media, B.V., 2001

work page 2001

[12] [12]

Wan and W

J. Wan and W. Zudilin, Generating functions of Legendre polynomials: a tribute to Fred Brafman, J. Approx. Theory 164 (2012), 488–503. Independent Researcher, 76520 Franqueville-Saint-Pierre, France Email address:michelbataille@wanadoo.fr Independent Researcher, 72764 Reutlingen, Germany Email address:robert.frontczak@web.de

work page 2012