Power series for roots of a trinomial and Kummer-like identities for higher order hypergeometric series

S. R. Mane

arxiv: 2606.23750 · v1 · pith:LDVGBY5Gnew · submitted 2026-06-22 · 🧮 math.CO

Power series for roots of a trinomial and Kummer-like identities for higher order hypergeometric series

S. R. Mane This is my paper

Pith reviewed 2026-06-26 08:28 UTC · model grok-4.3

classification 🧮 math.CO

keywords trinomial equationhypergeometric seriesKummer identitiesdiscriminantpower seriesroots of polynomialsalgebraic equations

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The pith

Roots of x^n + p x + q =0 admit hypergeometric series in powers of the discriminant D for all n at least 3.

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

The paper begins from known series solutions for the roots of the trinomial x^n + p x + q =0 expressed in powers of q^{n-1}/p^n. For the cubic case it recovers alternative expansions in the discriminant D and in 1/D by applying Kummer identities. It then constructs corresponding series in D and 1/D that hold for every n greater than or equal to 3. These new expansions indicate the existence of summation identities for hypergeometric series of order higher than two that parallel the classical Kummer relations.

Core claim

For the equation x^n + p x + q =0 with real nonzero p and q and integer n at least 3, the roots admit hypergeometric series representations in powers of the discriminant D = (n-1)^{n-1} (-p)^n - n^n q^{n-1} (and likewise in powers of 1/D). When n=3 these series coincide with those obtained from Kummer identities; for larger n they furnish evidence that analogous identities exist among higher-order hypergeometric functions.

What carries the argument

Hypergeometric series for the roots written in powers of the trinomial discriminant D.

If this is right

Roots of the trinomial can be written as D-power series for every n greater than or equal to 3.
The same roots also possess reciprocal-D series expansions.
The cubic series recovered via Kummer identities are recovered as the n=3 special case.
The pattern of the new series points to summation relations among hypergeometric functions of order n-1 or higher.

Where Pith is reading between the lines

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

If the suggested identities hold, closed-form reductions may become available for certain hypergeometric series of arbitrary order.
The discriminant-based expansions could be tested on other depressed polynomials whose discriminants are known explicitly.
Numerical verification for small n and moderate |D| would provide immediate evidence for or against the higher-order identities.

Load-bearing premise

The newly constructed hypergeometric series in powers of D converge to the true roots inside the parameter ranges given for each n at least 3.

What would settle it

Pick concrete values such as n=4, p=-1, q=1, compute the four numerical roots to high precision, then sum the first twenty terms of the proposed D-series and check whether the partial sums approach those roots.

read the original abstract

We study the trinomial equation $x^n +px +q =0$. Here $p$ and $q$ are both real and nonzero. For $n\ge3$, expressions for the roots have been published as hypergeometric series in powers of the parameter $q^{n-1}/p^n$. For the special case of the cubic ($n=3$), we employ Kummer's identities to derive alternative series solutions in powers of the discriminant $D$, and also series in powers of $1/D$. We next derive new series, in powers of $D$ and also in powers of $1/D$, for all $n\ge 3$. The resulting series suggest identities analogous to Kummer's identities, for higher order hypergeometric series.

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 writes down formal hypergeometric series in the discriminant D for roots of x^n + p x + q =0 when n>3, but supplies no check that those series actually satisfy the polynomial.

read the letter

The central point is that this work extends the cubic case by producing series in powers of D and 1/D for the roots of the trinomial when n is at least 4. For n=3 it applies Kummer identities to known _2F1 solutions to obtain the alternative expansions, which is a straightforward and correct step. For higher n it presents analogous formal series and suggests they point to new identities for higher-order hypergeometric functions.

The derivations for the cubic look clean and rest on established identities. The higher-n expressions are written out explicitly in the same style, which gives a concrete object that specialists can examine.

The soft spot is the missing verification step. The paper does not substitute the new series back into the original equation to confirm they are roots, nor does it give a convergence argument that would show the series represent the actual algebraic roots inside the claimed parameter ranges rather than formal solutions to a different equation. For n=3 this is covered by the Kummer identities; for n>3 the claim rests on the formal analogy alone. That gap is the main limitation.

The citation pattern follows the usual references for trinomial roots and hypergeometric identities, with no obvious circularity. The math is presented as direct manipulation of series coefficients.

This paper is for people who work on explicit root formulas and special-function identities in algebra and combinatorics. A reader already familiar with the cubic case and looking for higher-degree extensions would get the most out of the explicit series expressions, even if the verification is left open.

I would send it to peer review. The formal series are worth a referee's time to check the coefficient derivations and to see whether the suggested identities can be made rigorous.

Referee Report

2 major / 2 minor

Summary. The manuscript studies roots of the trinomial x^n + p x + q =0 (p,q real, nonzero) for n≥3. Existing hypergeometric series in q^{n-1}/p^n are recalled; for n=3, Kummer identities are used to obtain alternative series in the discriminant D and in 1/D. New hypergeometric series in D and 1/D are then derived for general n≥3, and these are said to suggest Kummer-like identities for higher-order hypergeometric series.

Significance. If the series are shown to converge to the actual roots, the work would supply new explicit representations for trinomial roots and indicate possible extensions of Kummer's theorem to _ {n-1}F_{n-2} series. The formal derivations for n>3 constitute the main technical contribution.

major comments (2)

[Derivation of new series for n≥3] The derivation of the D-powered series for n>3 (the paragraph beginning 'We next derive new series... for all n≥3') supplies formal hypergeometric expressions but contains no verification that these series satisfy the original algebraic equation x^n + p x + q =0, nor any radius-of-convergence estimate guaranteeing they equal the roots inside the stated parameter ranges. For n=3 the claim rests on known Kummer identities; for n>3 this step is load-bearing and missing.
[Results for general n] No independent check (substitution back into the trinomial, numerical evaluation, or comparison with the known q^{n-1}/p^n series) is reported for the new D-series when n≥4. Without such a check the claim that the series represent the roots remains an unproven formal identity.

minor comments (2)

[Abstract] The abstract states that the series 'suggest identities'; the manuscript should explicitly state whether these identities are proven or remain conjectural.
[Introduction and notation] Notation for the discriminant D and the precise parameter ranges should be restated uniformly before the general-n derivations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, acknowledging that the derivations for n>3 are formal and that the manuscript would benefit from explicit clarification on their status. We propose partial revisions to strengthen the presentation without altering the core contribution.

read point-by-point responses

Referee: The derivation of the D-powered series for n>3 (the paragraph beginning 'We next derive new series... for all n≥3') supplies formal hypergeometric expressions but contains no verification that these series satisfy the original algebraic equation x^n + p x + q =0, nor any radius-of-convergence estimate guaranteeing they equal the roots inside the stated parameter ranges. For n=3 the claim rests on known Kummer identities; for n>3 this step is load-bearing and missing.

Authors: We agree that the derivation for n>3 provides formal hypergeometric expressions obtained by generalizing the term structure from the n=3 case (via the known q^{n-1}/p^n series and the discriminant parameterization) but does not include direct substitution verification or radius-of-convergence estimates. The manuscript frames these as suggesting Kummer-like identities for _{n-1}F_{n-2} series rather than as proven representations of the roots. We will revise the text in the relevant paragraph and the abstract to state explicitly that the series are formally derived and that equality to the roots for n>3 is conjectural, pending a general extension of Kummer's theorem. revision: partial
Referee: No independent check (substitution back into the trinomial, numerical evaluation, or comparison with the known q^{n-1}/p^n series) is reported for the new D-series when n≥4. Without such a check the claim that the series represent the roots remains an unproven formal identity.

Authors: The manuscript reports no such independent checks for n≥4, consistent with its emphasis on formal derivation and the suggestion of new identities rather than a complete proof. We acknowledge that this leaves the representation of the roots as an unproven formal identity for general n. In revision we will add a clarifying sentence noting the formal character of the claim for n>3 and, if space allows, include a short numerical comparison for the n=4 case against the known q-series to illustrate consistency. revision: partial

Circularity Check

0 steps flagged

Derivation chain self-contained; no reductions by construction

full rationale

The paper applies established external Kummer identities to obtain D-powered series for the cubic case (n=3), then states that it derives new formal series in D and 1/D for n>=3. No quoted step defines a quantity in terms of the target result, renames a fitted parameter as a prediction, or relies on a self-citation chain whose content is unverified. The suggestion of higher-order Kummer-like identities is presented as a consequence of the explicit series constructions rather than an input assumption. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities.

pith-pipeline@v0.9.1-grok · 5654 in / 972 out tokens · 17359 ms · 2026-06-26T08:28:37.191243+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references

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R. Birkeland, Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen.Math- ematische Zeitschrift,26(1927) 566–578

1927

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J. Lambert, “Observations analytiques,”Nouveaux Mémoires de l’Académie Royale des Sciences et Belles- Lettres, Berlin, (1770) pp. 225–244

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S. R. Mane, Multiparameter Fuss-Catalan numbers with application to algebraic equations. arXiv:2410.07922 [math.CO], (2016)

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