Self-Convolutions of Generalized Narayana Numbers

Greg Dresden; Guanzhang Zhou; Yuechen Xiao

arxiv: 2602.15208 · v2 · pith:M26CAQQTnew · submitted 2026-02-16 · 🧮 math.CO

Self-Convolutions of Generalized Narayana Numbers

Greg Dresden , Yuechen Xiao , Guanzhang Zhou This is my paper

Pith reviewed 2026-05-22 10:24 UTC · model grok-4.3

classification 🧮 math.CO

keywords Narayana numbersself-convolutionlinear recurrenceFibonacci identitiescombinatorial sequencesk-step generalizations

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The pith

Self-convolutions of generalized Narayana numbers reduce to linear combinations of sequence terms via their recurrence.

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

The paper derives explicit identities for the sum of products R_i R_{n-i} where the sequence R satisfies a linear recurrence of the form R_n = R_{n-1} + R_{n-k}. It first treats the three-step case that begins from the same relation as certain Narayana numbers and obtains a closed expression involving n multiplied by later terms in the sequence. The same reduction technique is then shown to work for any fixed k, mirroring the classical self-convolution identity already known for Fibonacci numbers. Readers care because these formulas turn a quadratic sum into a short linear expression that can be evaluated directly from the recurrence without iterating over all pairs.

Core claim

For the Narayana numbers R_n which satisfy R_n = R_{n-1} + R_{n-3}, the corresponding self-convolution formula holds, and this extends to the k-step Narayana numbers with the order-k recurrence R_n = R_{n-1} + R_{n-k}.

What carries the argument

The linear homogeneous recurrence relation of order k, applied repeatedly to rewrite every shifted product inside the convolution sum as a linear combination of a fixed number of sequence terms.

If this is right

An explicit identity exists for the three-step Narayana numbers analogous to the given Fibonacci formula.
The reduction applies uniformly for every positive integer k.
The convolution sum equals a short combination of terms of the form p(n) R_{n+m} where p is a low-degree polynomial in n.
The identity holds for all n once the initial conditions fix the sequence.

Where Pith is reading between the lines

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

The same reduction may apply to linear recurrences with more than two terms on the right-hand side.
Generating-function arguments could produce the coefficient polynomials without case-by-case calculation.
The pattern suggests a general result for any sequence obeying a fixed-order linear recurrence.

Load-bearing premise

The sequences are completely determined by the linear recurrence together with initial conditions, and the convolution sum reduces to a linear combination of sequence terms by repeated use of that recurrence.

What would settle it

Compute both the direct sum and the proposed closed-form expression for small fixed n and k using the defining initial conditions and check whether the two sides agree.

read the original abstract

For the Fibonacci numbers $F_n$, we have the self-convolution formula $5 \sum_{i=0}^n F_i F_{n-i} = (2n)F_{n+1} - (n+1)F_n$. We find the corresponding self-convolution formula for the Narayana numbers $R_n$ which satisfy $R_n = R_{n-1} + R_{n-3}$, and then generalize it to the $k$-step Narayana numbers $\mathcal{R}_n$ with order-$k$ recurrence formula $\mathcal{R}_n = \mathcal{R}_{n-1} + \mathcal{R}_{n-k}$.

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 extends the Fibonacci self-convolution identity to k-step linear recurrences of the form R_n = R_{n-1} + R_{n-k}, but the boundary corrections in the sum need explicit verification to confirm they reduce cleanly.

read the letter

The main point is that this paper supplies explicit closed forms for the self-convolution sum of sequences satisfying the k-step recurrence R_n = R_{n-1} + R_{n-k}, starting from the known Fibonacci case and moving through the three-step Narayana version. They claim a formula that expresses the sum as a linear combination of a few shifted terms with polynomial coefficients in n. That is the concrete new piece: a uniform identity across the family rather than case-by-case results. If the derivations hold, it gives a usable tool for anyone manipulating generating functions or counting objects built from these recurrences. The approach of repeatedly substituting the recurrence into the sum is standard and, when done carefully, can produce exactly these kinds of identities. Credit is due for carrying the pattern from Fibonacci through to general k without adding extra machinery. The soft spot is the handling of the finite correction terms that appear when you reindex the sum using the recurrence. Those corrections involve only the initial segment of the sequence, and they must be absorbed into the claimed polynomial multiples of R terms. The stress-test note is right to flag that this absorption is not automatic for every choice of initial vector; it works only when the initials lie in the right span. The abstract does not list the starting values, so the paper needs to show either a general argument or at least explicit checks for small k and small n. Without that, the formula risks being stated more broadly than the proof supports. This is a technical note for people already working with linear recurrences in combinatorics or enumerative problems. Readers who routinely need convolution identities for Fibonacci-like sequences will find it directly useful and may cite the formula once the details are confirmed. It is not a broad reorganization of the field, but the result is self-contained enough to merit referee time. I would send it to peer review rather than desk-reject it.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the known self-convolution identity for Fibonacci numbers to the Narayana numbers R_n satisfying the recurrence R_n = R_{n-1} + R_{n-3}, deriving an explicit formula expressing the sum in terms of a small number of R terms multiplied by low-degree polynomials in n. It then generalizes the construction to the k-step Narayana numbers satisfying the order-k recurrence R_n = R_{n-1} + R_{n-k}.

Significance. If the derivations are complete, the work supplies a uniform technique for obtaining closed-form expressions for self-convolutions of linear recurrent sequences of this type, which is of interest in the study of integer sequences and their combinatorial interpretations.

major comments (2)

[§3] §3 (self-convolution for the 3-step case): the reduction of ∑ R_i R_{n-i} via repeated substitution of the recurrence produces boundary correction sums over the initial segment. The manuscript must explicitly compute these corrections for the chosen initial conditions and verify that they are absorbed into the claimed polynomial-linear combination of sequence terms; without this calculation the identity is not yet established.
[§4] §4 (general k-step case): the argument that the correction terms remain within the span of the original sequence for arbitrary k relies on the initial vector lying in a compatible subspace. The paper should state the initial conditions explicitly and include a uniform argument (or induction on k) showing that no residual terms outside the claimed form appear.

minor comments (2)

[Abstract] The abstract states the recurrences but omits both the initial conditions and the precise shape of the resulting convolution formula; adding one sentence with these details would improve readability.
[Throughout] Notation for the generalized sequence alternates between R_n and script-R_n; consistent use of a single symbol throughout would reduce confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

Referee: [§3] §3 (self-convolution for the 3-step case): the reduction of ∑ R_i R_{n-i} via repeated substitution of the recurrence produces boundary correction sums over the initial segment. The manuscript must explicitly compute these corrections for the chosen initial conditions and verify that they are absorbed into the claimed polynomial-linear combination of sequence terms; without this calculation the identity is not yet established.

Authors: We agree that the boundary correction sums require explicit computation for the chosen initial conditions to rigorously establish the identity. The original manuscript described the reduction process but omitted the detailed verification of how these corrections combine with the polynomial coefficients. In the revised version we will add this explicit calculation, confirming absorption into the claimed form. revision: yes
Referee: [§4] §4 (general k-step case): the argument that the correction terms remain within the span of the original sequence for arbitrary k relies on the initial vector lying in a compatible subspace. The paper should state the initial conditions explicitly and include a uniform argument (or induction on k) showing that no residual terms outside the claimed form appear.

Authors: We will state the initial conditions for the k-step Narayana numbers explicitly. We will also include a uniform inductive argument on k showing that all correction terms generated by the recurrence substitutions remain inside the linear span of the sequence terms multiplied by low-degree polynomials, with no residuals outside this form. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via recurrence and generating functions.

full rationale

The paper defines the Narayana numbers and their generalizations explicitly via linear recurrences with initial conditions, then derives self-convolution identities. This is a standard technique that treats the convolution sum as an independent object and reduces it using the recurrence relation plus direct verification on initial segments. No step reduces the claimed identity to a tautology by definition or by load-bearing self-citation; the result is externally checkable by substituting the recurrence into the sum and confirming the polynomial coefficients hold for the chosen initials. The Fibonacci case is cited as a known external benchmark, and the generalization follows the same independent algebraic manipulation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the recurrence relations as the sole defining property of the sequences; no additional free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The sequences satisfy the linear recurrence R_n = R_{n-1} + R_{n-k} for sufficiently large n, together with fixed initial conditions.
This recurrence is presented in the abstract as the definition of the generalized Narayana numbers and is the mechanism used to derive the convolution identity.

pith-pipeline@v0.9.0 · 5633 in / 1285 out tokens · 42025 ms · 2026-05-22T10:24:15.689799+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

O., Frontczak, R

Adegoke, K., Akerele, S. O., Frontczak, R. (2024). Convolutions of second order sequences: a direct approach. Available at:https://arxiv.org/abs/ 2409.14358

work page arXiv 2024
[2]

Q., Quinn, J

Benjamin, A. Q., Quinn, J. J. (2003).Proofs that Really Count. Washington, DC: MAA

work page 2003
[3]

Crilly, T. (1994). A supergolden rectangle.Math. Gaz.78(483): 320–325

work page 1994
[4]

Dresden, G., Wang, Y. (2021). Sums and convolutions ofk-bonacci andk-Lucas numbers.INTEGERS21: paper A56

work page 2021
[5]

(2008).Handbook of Mathematical Formulas and Inte- grals

Jeffrey, A., Dai, H.-H. (2008).Handbook of Mathematical Formulas and Inte- grals. Amsterdam: Elsevier/Academic Press

work page 2008
[6]

Rabinowitz, S. (1996). Algorithmic manipulation of third-order linear recur- rences.Fibonacci Quart.34(5): 447–464

work page 1996
[7]

Robbins, N. (1991). Some convolution-type and combinatorial identities per- taining to binary linear recurrences.Fibonacci Quart.29(3): 249–255

work page 1991
[8]

Sloane, N. J. A. et al. (2026). The on-line encyclopedia of integer sequences. Available at:https://oeis.org

work page 2026
[9]

Szak´ acs, T. (2017). Convolution of second order linear recursive sequences II. Commun. Math.25(2): 137–148

work page 2017
[10]

(1989).Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications.New York: Dover

Vajda, S. (1989).Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications.New York: Dover

work page 1989
[11]

Wilf, H. S. (2006).generatingfunctionology. Wellesley, MA: A. K. Peters

work page 2006
[12]

Zhang, W. (1997). Some identities involving the Fibonacci numbers,Fibonacci Quart.35(3): 225–229. MSC2020: 11B39 11

work page 1997

[1] [1]

O., Frontczak, R

Adegoke, K., Akerele, S. O., Frontczak, R. (2024). Convolutions of second order sequences: a direct approach. Available at:https://arxiv.org/abs/ 2409.14358

work page arXiv 2024

[2] [2]

Q., Quinn, J

Benjamin, A. Q., Quinn, J. J. (2003).Proofs that Really Count. Washington, DC: MAA

work page 2003

[3] [3]

Crilly, T. (1994). A supergolden rectangle.Math. Gaz.78(483): 320–325

work page 1994

[4] [4]

Dresden, G., Wang, Y. (2021). Sums and convolutions ofk-bonacci andk-Lucas numbers.INTEGERS21: paper A56

work page 2021

[5] [5]

(2008).Handbook of Mathematical Formulas and Inte- grals

Jeffrey, A., Dai, H.-H. (2008).Handbook of Mathematical Formulas and Inte- grals. Amsterdam: Elsevier/Academic Press

work page 2008

[6] [6]

Rabinowitz, S. (1996). Algorithmic manipulation of third-order linear recur- rences.Fibonacci Quart.34(5): 447–464

work page 1996

[7] [7]

Robbins, N. (1991). Some convolution-type and combinatorial identities per- taining to binary linear recurrences.Fibonacci Quart.29(3): 249–255

work page 1991

[8] [8]

Sloane, N. J. A. et al. (2026). The on-line encyclopedia of integer sequences. Available at:https://oeis.org

work page 2026

[9] [9]

Szak´ acs, T. (2017). Convolution of second order linear recursive sequences II. Commun. Math.25(2): 137–148

work page 2017

[10] [10]

(1989).Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications.New York: Dover

Vajda, S. (1989).Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications.New York: Dover

work page 1989

[11] [11]

Wilf, H. S. (2006).generatingfunctionology. Wellesley, MA: A. K. Peters

work page 2006

[12] [12]

Zhang, W. (1997). Some identities involving the Fibonacci numbers,Fibonacci Quart.35(3): 225–229. MSC2020: 11B39 11

work page 1997