Sums of powers over equally spaced Fibonacci numbers

Helmut Prodinger

arxiv: 1907.06103 · v2 · pith:KP5KLCHVnew · submitted 2019-07-13 · 🧮 math.NT

Sums of powers over equally spaced Fibonacci numbers

Helmut Prodinger This is my paper

Pith reviewed 2026-05-24 21:54 UTC · model grok-4.3

classification 🧮 math.NT

keywords Fibonacci numberssums of powersequally spaced indicesclosed-form sumsgenerating functionsarithmetic progressions

0 comments

The pith

Formulas for sums of cubes of Fibonacci numbers extend directly to sums of arbitrary powers.

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

The paper takes existing closed-form results for the sum of cubes of Fibonacci numbers in arithmetic progression and shows that the same identities and derivations continue to hold when the exponent is any positive integer. It demonstrates this by applying the prior algebraic and generating-function methods without requiring new conditions or case distinctions. A sympathetic reader would care because the extension supplies uniform expressions that let one compute or simplify high-power Fibonacci sums in a single framework rather than handling each exponent separately. The central objects are the partial sums over every mth Fibonacci number raised to the kth power, for fixed m and variable k.

Core claim

The summation identities previously established for the third powers of equally spaced Fibonacci numbers remain valid when the exponent is replaced by an arbitrary positive integer k; the generating-function or recurrence-based proofs carry over verbatim.

What carries the argument

Generating functions or algebraic identities that encode the sums of kth powers of Fibonacci numbers with indices in arithmetic progression.

If this is right

Closed-form expressions exist for the sum of kth powers for every positive integer k.
The same recurrence or generating-function setup produces the formula uniformly in k.
No additional identities are needed when the exponent increases beyond three.
The resulting expressions remain linear combinations of Fibonacci numbers and Lucas numbers or similar companion sequences.

Where Pith is reading between the lines

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

The same direct extension may apply to other linear recurrence sequences such as Lucas or Pell numbers.
One could test whether the formulas remain valid when the common difference m is allowed to vary with k.
The uniform treatment might simplify proofs of congruences or divisibility properties for higher-power Fibonacci sums.
Numerical verification for several small k would provide an immediate check on the claimed generality.

Load-bearing premise

The algebraic or generating-function techniques that worked for the specific case of cubes extend directly and without modification or extra conditions to the case of arbitrary powers.

What would settle it

Explicit computation of the sum for k=4 and a small arithmetic progression that fails to match the closed form obtained by the cube-case method.

read the original abstract

Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.

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

Prodinger extends the 2018 cube-sum identity for Fibonacci numbers to arbitrary powers via the same algebraic approach.

read the letter

Prodinger's paper extends the 2018 Frontczak result on sums of cubes of Fibonacci numbers to arbitrary powers, still for equally spaced terms. That's the main new thing here. The paper carries over the algebraic or generating-function method from the cube case to the general power case. This seems to work without modification or extra conditions, which is a positive. The title and abstract indicate a direct extension, and the citation pattern looks appropriate. The limitation is that this remains an incremental contribution in a narrow corner of number theory. It does not resolve open questions or provide a new framework. The abstract gives no details on the derivation or verification, so the full paper must supply those to make the claim solid. No load-bearing flaws are visible from the given information. This is the sort of paper that specialists in Fibonacci identities will appreciate for the explicit results it provides. A reader working on similar sums would get some value from the closed forms or relations derived. I would send this to peer review. The work is grounded in prior results and the extension is precise enough to warrant checking the details by referees. The stress test note aligns with this view that there is no significant objection at this level.

Referee Report

0 major / 2 minor

Summary. The manuscript extends results from Frontczak (2018) on sums of cubes of Fibonacci numbers to sums of arbitrary powers, specifically over equally spaced terms in the Fibonacci sequence, using algebraic or generating-function techniques.

Significance. If the derivations hold, the work supplies a general closed-form treatment for higher-power sums, strengthening the literature on Fibonacci identities with a direct, parameter-free extension of prior cube-sum formulas. This is a standard but useful move in the field when the algebraic steps are fully rigorous.

minor comments (2)

[Abstract] Abstract: the single-sentence claim provides no indication of the method, the form of the resulting identity, or any verification; a slightly expanded abstract would better convey the contribution.
The reference to Frontczak (2018) is given but the manuscript should explicitly state which specific cube-sum identity is being generalized (e.g., the exact formula being extended).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends prior results on sums of cubes of Fibonacci numbers (cited to external author Frontczak 2018) to arbitrary powers via algebraic or generating-function methods. No self-citations are load-bearing, no parameters are fitted and renamed as predictions, and no derivation steps reduce by construction to inputs. The work is a standard direct extension in the Fibonacci identities literature and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities; ledger is empty by necessity.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean (washburn_uniqueness_aczel, phi_fixed_point); Constants.lean (phi_golden_ratio) phi_golden_ratio; Jcost uniqueness echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Binet formulæ: √5 F_n = α^n − β^n ... α = (1+√5)/2, β=(1−√5)/2; expansions F_n^j = (1/5^{j/2}) ∑ ... F_{(j−2s)n} ... and analogous Lucas forms; sums derived from generating functions ∑ F_{mn} z^n = z F_m / (1 − L_m z + (−1)^m z^2)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.