A Short Proof of Knuth's Old Sum

Kunle Adegoke

REVIEW 1 major objections 5 minor 12 references

A Short Proof of Knuth's Old Sum

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Knuth's old sum is proved by substituting $y=-\cos x-1$ into the binomial theorem and integrating termwise with Beta-function integrals.

desk verdict Correct short proof of Knuth's old sum; the v-generalizations overstate their domain and need a fix before they can be trusted. read the letter →

arxiv 2412.00040 v2 pith:MTK3NB5N submitted 2024-11-22 math.GM

Kunle Adegoke This is my paper · ORCID

classification math.GM MSC 05A1005A19

keywords Knuth'soldsumReedDawsonidentityBetafunctionbinomialtheoremgeneralizedcoefficientsCatalannumberscombinatorialidentitiespolynomial

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper gives a short proof of Knuth's old sum, the identity $\sum_{k=0}^n (-1)^k \binom{n}{k} 2^{-k}\binom{2k}{k} = 2^{-n}\binom{n}{n/2}$ for even $n$ and $0$ for odd $n$. The proof substitutes $y=-\cos x-1$ into the ordinary binomial theorem and integrates termwise from $0$ to $\pi$; every integral is evaluated through Beta-function formulas for powers of half-angle sines and cosines. The same integration step is then applied to a two-parameter family and to a general scheme that converts any polynomial identity of a certain form into combinatorial identities. If the identities are correct, the paper supplies a unified derivation of a known summatory result and a small library of new binomial and Catalan-number identities.

What carries the argument

The load-bearing object is the Beta-function evaluation of trigonometric integrals in Lemma 1, especially $I(u,v)=\int_0^\pi \cos^u(x/2)\sin^v(x/2)\,dx = 2^{-u-v}\pi\binom{u}{u/2}\binom{v}{v/2}\binom{(u+v)/2}{u/2}^{-1}$, together with the fact that $\int_0^\pi \cos^m x\,dx=0$ for odd integer $m$. Substituting $y=-\cos x-1$ into the binomial theorem converts the alternating sum into an integral of a power of cosine, and termwise integration collapses via these formulas. The same machinery, formalized in Theorems 9-13, turns any polynomial identity of the displayed form into weighted sums by integrating against Beta kernels.

What would settle it

Evaluate (4.2) with $n=2$ and $v=-2$: the $k=2$ summand contains $\binom{0}{-1}^{-1}$, which is not finite, so the claimed equality for all real $v$ breaks down as a finite identity; for the original Knuth sum itself, $n=4$ gives $3/8$ on both sides and $n=5$ gives $0=0$, so the two claims separate cleanly.

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Extended reading notes

Core claim

The central claim is that Knuth's old sum follows from one identity: after the substitution $y=-\cos x-1$, the binomial theorem reads $\sum_{k=0}^n (-1)^k\binom{n}{k}2^k\cos^{2k}(x/2)=(-1)^n\cos^n x$, and termwise integration combined with $\int_0^\pi \cos^u(x/2)\,dx=2^{-u}\pi\binom{u}{u/2}$ and the vanishing of $\int_0^\pi \cos^n x\,dx$ for odd $n$ yields (1.1) in a few lines. The paper further claims a two-parameter generalization, identity (4.1), in which a second non-negative integer $m$ and a real parameter $v$ are inserted through multiplication by $(1+\cos x)^m\sin^v x$; Corollary 3 isolates the $m=0$ case as a $v$-deformation of Knuth's old sum. Finally, the paper claims that the same Beta-integration technique, applied to a generic polynomial identity of the form $\sum_k f(k)(1+t)^{p(k)}=\sum_k g(k)t^{q(k)}$, generates families of polynomial identities and combinatorial identities, including complements of Knuth's old sum and Catalan-number identities.

Load-bearing premise

The generalized theorems say $v$ can be any real number, but the integral evaluations used to prove them are derived only for $\Re v>-1$, and no analytic-continuation argument is supplied to close the gap.

Editorial extensions

If this is right

The alternating sum (1.1) is closed: even $n$ gives $2^{-n}\binom{n}{n/2}$ and odd $n$ gives $0$.
The two-parameter identity (4.1) generalizes (1.1); its $v$-deformation (4.2) and the $m$-only version (1.2) follow as special cases.
Setting $x=0$ and $x=1$ in the polynomial identity (1.6) recovers Knuth's old sum and the Riordan special case (1.3), so that polynomial identity subsumes both.
The complements in Section 5 yield the classical convolution $\sum_{k=0}^n \binom{2(n-k)}{n-k}\binom{2k}{k}=2^{2n}$ and a signed counterpart that vanishes for odd $n$.
The Beta-integration machinery produces fresh binomial and Catalan-number identities, for instance $\sum_{k=1}^{\lceil n/2\rceil}\binom{n}{2k-1}2^{n-2k}C_k = \tfrac12 C_{n+2}-C_{n+1}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same Beta-kernel integration should apply to the $q$-binomial theorem, yielding $q$-analogues of Knuth's old sum; the paper does not pursue this.
Evaluating the polynomial identities at algebraic points other than $0,1,2,-1$ would produce further closed-form binomial sums along the same template.
A fair reading of the method's domain is $\Re v>-1$; extending the $v$-generalizations to all real $v$ would require a separate analytic-continuation argument.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The proof of Knuth's old sum is correct, and it is genuinely short: substitute y = -cos x - 1 into the binomial theorem, integrate, and evaluate with Beta integrals. That is the paper's real contribution, and it is solid. The broader package—the v-parameter generalizations, the polynomial identities in Sections 7 and 8, and the Waring/Simons corollaries—contains a lot that looks new, and the derivations are transparent enough to check. Spot checks of Theorem 2, Corollary 3, and several Section 7 identities at small n gave the stated values.

The soft spot is exactly where the stress-test note lands. Lemma 1 needs Re u > -1 and Re v > -1. Theorem 2, Corollary 3, and about a dozen later statements claim v is any real number. The proofs never supply the analytic-continuation argument that would extend the identities to v <= -1, and for many negative v the integrands are not integrable or the generalized binomial coefficients are undefined. So the generalizations are proved only for v > -1, and the stated domain is unsupported. This is a genuine rigor gap, but it is fixable: restrict the statements to v > -1, or spell out the continuation and the excluded values. It does not touch the central identity, where v = 0.

Also worth fixing: the proof of Theorem 25 has a typo in the displayed equation after the binomial expansion (the 'cos^{2n-k} x sin^k x' line should involve half-angles), and the paper would be easier to use if the author explicitly separated what is new in Theorem 2 from the recent generalizations in [8, 11, 5, 2, 1]. Those are minor.

Bottom line: this is a useful paper for anyone working with Knuth-sum type identities. The main proof is elegant, the machinery is reproducible, and the domain gap is not load-bearing for the headline result. Send it to a referee, with a request that the author fix the v-domain statements and the typo. I would not cite the v-generalizations until that fix is in.

Referee Report

1 major / 5 minor

Summary. The paper presents a short proof of Knuth's old sum (1.1) by substituting y = -cos x - 1 into the binomial theorem, integrating over [0, pi], and evaluating the resulting Beta-function integrals. It then develops a two-parameter generalization (Theorem 2, Eq. (4.1)), a v-parameter generalization (Corollary 3), complements (Theorems 5-8), a general transformation machine for polynomial identities (Section 6), applications to the binomial theorem and Waring's formulas (Sections 7-8), and many combinatorial corollaries. The proof of (1.1) is correct and short; the generalizations are derived by the same integration technique. The principal weakness is that the generalization theorems state 'v is a real number' while the proofs rely on Lemma 1, which requires Re v > -1, and no analytic-continuation argument is given for v <= -1.

Significance. The central proof of (1.1) is a genuine, clean contribution: it reduces the identity to the binomial theorem and two Beta-function integrals, with no fitted parameters and no circular appeal to the target identity. The paper also demonstrates a systematic method for producing many related identities, and spot checks of Theorem 2 and Corollary 3 at small n and v are consistent. If the domain restrictions are corrected, the generalization framework appears sound and the paper would be a useful compendium of identities. As written, however, the unqualified 'v real' statements materially overstate the established range, so the paper needs revision before it can be accepted.

major comments (1)

[Sections 4-8, esp. Theorem 2 (Eq. (4.1)) and Corollary 3 (Eq. (4.2))] The statements of Theorem 2, Corollary 3, Corollary 4, Theorem 7, Theorem 8, Theorem 16, Theorem 20, Theorems 23-26, and Propositions 7-8 assert that v is a real number, but every proof uses Lemma 1, whose integral evaluations (2.2)-(2.5) require Re v > -1 (and Re u > -1). For v <= -1, the kernel sin^v(x/2) is not integrable at x = 0, and for negative-integer v the generalized binomial coefficients in (2.1) have poles, so the summands are not even defined in general. No meromorphic-continuation argument or exclusion of singular v is supplied. The proofs therefore establish these identities only for v > -1, and the 'v real' claims are unsupported. This does not affect Theorem 1 or identity (1.1), where v = 0 lies inside the valid domain, but it is a load-bearing gap in the advertised generalizations.

minor comments (5)

[Section 8.1, proof of Theorem 25] The line 'Since (cos(x/2)+sin(x/2))^{2m} = (1+sin x)^m' introduces an undefined parameter m; the subsequent expansion uses 2n, so the exponent should be 2n and the right-hand side should be (1+sin x)^n.
[Section 1, after Eq. (1.2)] The statements that (1.3) and (1.4) are obtained by setting n = 0 and n = 1 in (1.2) are imprecise: (1.3) requires additionally multiplying by 2^n, and (1.4) requires multiplying the n = 1 case by -2^{n+1} (or an equivalent normalization). Please state the exact specialization.
[Section 2, first paragraph] The phrase 'the rest of the complex plain' should read 'the rest of the complex plane'.
[Reference [7]] The journal name 'EACTS Bulletin' appears to be a typo for 'EATCS Bulletin'.
[Theorem 10] The hypothesis 'for every non-negative integer j' should quantify over the indices k in the finite summation ranges, since p(k) and q(k) are only defined there.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main Knuth-sum proof reduces to the binomial theorem plus independent Beta-function integrals; the generalizations contain a domain-validity gap ('v real' versus Lemma 1's Re v > -1), but that is a rigor issue, not circular reasoning.

full rationale

The derivation chain for the central identity (1.1) is self-contained and non-circular. Theorem 1 substitutes y = -cos x - 1 into the binomial theorem to obtain Eq. (3.1), integrates term by term, and evaluates the resulting integrals using Lemma 1, whose evaluations rest on the standard Beta-function integral quoted from Gradshteyn and Ryzhik. No term of the target sum is assumed, no parameter is fitted to the target identity, and no load-bearing self-citation enters: the only self-citation, Reference [1], is an editorial correction and is not used in any proof. The generalizations in Theorem 2, Corollary 3, Theorems 7, 8, 16, 20, 23-26, and related propositions are also derived constructively by multiplying Eq. (3.1) or its variants by powers of sine and cosine and integrating with Lemma 1, so they do not assume the identities they prove. The genuine weakness is a domain-validity gap, not circularity: Lemma 1 is stated only for Re u > -1 and Re v > -1, while many theorems state 'v is a real number' with no restriction; for v <= -1 the integral kernels sin^v(x/2) or sin^v x are not integrable at 0, and for negative-integer v the generalized binomial coefficients in (2.1) have poles. The paper supplies no meromorphic-continuation argument bridging this gap. Thus the proof as written establishes the generalized identities only on the domain where Lemma 1 applies, and the claims for all real v are unsupported. This is an honest limitation of the manuscript's stated domain, but it is not a reduction of any 'prediction' to its inputs. Since the main Knuth-sum identity is proved independently at v = 0, inside the valid domain, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proofs are self-contained derivations from the binomial theorem and standard Beta-function evaluations. The only external inputs are classical integral and Gamma identities, none of which encode the target sums.

assumptions (5)

standard math Beta integral formula K(u,v) = 2^{-u-v-1} pi C(u,u/2) C(v,v/2) C((u+v)/2,u/2)^{-1} from Gradshteyn-Ryzhik (2.6)
Foundations of Lemma 1 and all integral evaluations; cited from [4].
standard math Gamma duplication identities in (4.7) to (4.10)
Used to evaluate half-integer binomial coefficients in Corollaries and Propositions.
standard math Waring formula and its dual (8.20) and (8.21)
Used as source polynomial identities in Section 8.2; cited from [3].
standard math Simons identity (8.24)
Used as a source polynomial identity in Section 8.3; cited from [10].
standard math Definition of generalized binomial coefficients via Gamma functions (2.1)
Allows real parameter v; standard extension of binomial coefficients.

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Cite this review

Pith. "Pith review of A Short Proof of Knuth's Old Sum." pith.science (2026). https://pith.science/paper/MTK3NB5N

@misc{pith2026241200040,
  author       = {Pith},
  title        = {Pith review of: A Short Proof of Knuth's Old Sum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MTK3NB5N}},
  note         = {Machine review of arXiv:2412.00040}
}

read the original abstract

We give a short proof of the well-known Knuth's old sum and provide some generalizations. Our approach utilizes the binomial theorem and integration formulas derived using the Beta function. Several new polynomial identities and combinatorial identities are derived.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

K. Adegoke, Editorial Correction to ``A note on a generalization of Riordans combinatorial identity via a hypergeometric series approach'', Notes on Number Theory and Discrete Mathematics 30 :1 (2024), 211--212

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[2]

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[3]

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[4]

H. W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, The Fibonacci Quarterly 37 :2 (1999), 135--140

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Gradshteyn and I

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[7]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

K. Adegoke, Editorial Correction to ``A note on a generalization of Riordans combinatorial identity via a hypergeometric series approach'', Notes on Number Theory and Discrete Mathematics 30 :1 (2024), 211--212

work page 2024

[2] [2]

K. Adegoke, Editorial Correction to A note on a generalization of Riordan combinatorial identity via a hypergeometric series approach, Notes on Number Theory and Discrete Mathematics 30 :1 (2024), 211--212

work page 2024

[3] [3]

Alzer, Combinatorial identities and hypergeometric functions, II, Discrete Mathematics Letters 13 (2024), 1--5

H. Alzer, Combinatorial identities and hypergeometric functions, II, Discrete Mathematics Letters 13 (2024), 1--5

work page 2024

[4] [4]

H. W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, The Fibonacci Quarterly 37 :2 (1999), 135--140

work page 1999

[5] [5]

Gradshteyn and I

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, 7th edition, Elsevier Academic Press, Amsterdam, 2007

work page 2007

[6] [6]

Lim, A note on a generalization of Riordan combinatorial identity via a hypergeometric series approach, Notes on Number Theory and Discrete Mathematics 29 :3 (2023), 421--425

D. Lim, A note on a generalization of Riordan combinatorial identity via a hypergeometric series approach, Notes on Number Theory and Discrete Mathematics 29 :3 (2023), 421--425

work page 2023

[7] [7]

Miki\'c, A proof of a famous identity concerning the convolution of the central binomial coefficients, Journal of Integer Sequences 19 (2016), 1--10, Article 16.6.6

J. Miki\'c, A proof of a famous identity concerning the convolution of the central binomial coefficients, Journal of Integer Sequences 19 (2016), 1--10, Article 16.6.6

work page 2016

[8] [8]

Prodinger, Knuth old sum - a survey, EACTS Bulletin 52 (1994), 232--245

H. Prodinger, Knuth old sum - a survey, EACTS Bulletin 52 (1994), 232--245

work page 1994