Papers
Topics
Authors
Recent
Search
2000 character limit reached

Self-Convolutions of Generalized Narayana Numbers

Published 16 Feb 2026 in math.CO | (2602.15208v1)

Abstract: For the Fibonacci numbers FnF_n, we have the self-convolution formula 5∑i=0<sup>n</sup>FiFn−i=(2n)Fn+1−(n+1)Fn5 \sum_{i=0}<sup>n</sup> F_i F_{n-i} = (2n)F_{n+1} - (n+1)F_n. We find the corresponding self-convolution formula for the Narayana numbers RnR_n which satisfy Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}, and then generalize it to the kk-step Narayana numbers R<em>n\mathcal{R}<em>n with order-kk recurrence formula Rn=R</em>n−1+Rn−k\mathcal{R}_n = \mathcal{R}</em>{n-1} + \mathcal{R}_{n-k}.

Summary

  • The paper establishes explicit self-convolution formulas for k-step Narayana numbers, extending known results from Fibonacci and Lucas sequences.
  • The paper derives uniform convolution identities that link convolution coefficients to discriminants of characteristic polynomials using generating functions.
  • The paper demonstrates practical implications in combinatorial enumeration, algorithmic analysis, and potential applications in coding theory and discrete dynamics.

Self-Convolutions of Generalized Narayana Numbers

Introduction and Motivation

The paper addresses the self-convolution identities for higher-order recurrence sequences, with an emphasis on the Narayana numbers and their generalizations. While self-convolution formulas for the Fibonacci and Lucas numbers are well-established—such as 5∑i=0nFiFn−i=(2n)Fn+1−(n+1)Fn5 \sum_{i=0}^n F_i F_{n-i} = (2n)F_{n+1} - (n+1)F_n—there is a lack of systematic results for sequences defined by recurrences of order greater than two. The Narayana sequence, governed by Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}, extends the Fibonacci paradigm to third-order, prompting the study of its self-convolution formula and further generalizations.

Convolution Identities for Higher-Order Recurrences

The authors establish precise self-convolution formulas for sequences defined by kk-step recurrences, specifically examining and generalizing the Narayana numbers. For the Narayana numbers (k=3k=3), the convolution identity is:

31∑i=0nRiRn−i=9(n+1)Rn+2−3(n+3)Rn+1−2(n+2)Rn31\sum_{i=0}^n R_i R_{n-i} = 9(n+1)R_{n+2} - 3(n+3) R_{n+1} - 2(n+2)R_n

This result stems from a specialization of Rabinowitz's third-order recurrence convolution and is extended to the four-step Narayana sequence (Sn=Sn−1+Sn−4S_n = S_{n-1} + S_{n-4}):

283∑i=0nSiSn−i=64(n+2)Sn+3−16(n+5)Sn+2−12(n+4)Sn+1−9(n+3)Sn283\sum_{i=0}^n S_i S_{n-i} = 64(n+2)S_{n+3} - 16(n+5)S_{n+2} - 12(n+4)S_{n+1} - 9(n+3)S_n

The coefficients appearing in these formulas correspond to discriminants of characteristic polynomials for the recurrences (1−x−xk1-x-x^k), establishing a link between convolution structure and the underlying algebraic invariants.

Unified Generalization: kk-Step Narayana Numbers

The central theorem of the paper provides a uniform convolution formula for the kk-step Narayana numbers, Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}0, defined by:

Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}1

The self-convolution formula is:

Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}2

This result demonstrates an explicit structure for the convolution identities, governed by powers of Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}3 and Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}4, and provides a direct algorithm for convolution calculation in Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}5-step Narayana sequences. The proof utilizes generating function techniques and identities for finite geometric and arithmetic-geometric sums, with all coefficients interpreted through characteristic polynomial discriminants.

Numerical and Structural Properties

The paper highlights that for Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}6, the convolution coefficients (Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}7, Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}8, Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}9) are related to sums of powers and discriminants, specifically:

  • kk0
  • kk1
  • kk2

This correspondence is also reflected in the convolution formula structure for arbitrary kk3, establishing a combinatorial and algebraic connection useful for sequence analysis.

Theoretical and Practical Implications

The convolution identities derived serve as new tools for combinatorial enumeration and algebraic analysis of higher-order recurrence sequences. Practically, these formulas offer closed-form expressions for sequence convolutions, which is significant in combinatorial mathematics, algorithmic analysis, and potential applications in coding theory and discrete dynamics. Theoretically, the linkage between convolution structure, discriminants, and generating functions opens further avenues for research on generalized linear recurrence sequences and their algebraic properties.

Future Directions

Potential future developments include extensions of these convolution identities to more elaborate recurrence relations (e.g., with arbitrary coefficients), analysis of convolution behavior in non-homogeneous recurrences, and generalizations to multi-variable generating functions. Furthermore, exploration of the algebraic invariants underlying the convolution formulas—such as discriminants and characteristic roots—could yield deeper insight into sequence theory and combinatorial structures.

Conclusion

This paper systematically derives and generalizes self-convolution identities for the Narayana numbers and kk4-step analogues, framing the convolution structure through algebraic and combinatorial invariants. The results provide both explicit computational tools and theoretical frameworks for higher-order recurrence sequence analysis, strengthening the foundation for convolution studies beyond second-order recurrences (2602.15208).

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Explain it Like I'm 14

What is this paper about?

This paper looks at special number lists (called sequences) that grow by adding earlier terms. You might know one famous example: the Fibonacci numbers, where each number is the sum of the two before it. The authors study a similar family called the Narayana numbers and their generalizations. Their main goal is to find neat “shortcut” formulas for a kind of sum called a self‑convolution, which mixes pairs of numbers from the same sequence in a structured way.

The big questions

The paper answers simple questions like:

  • If you take a sequence that grows by “add the previous term and the one k steps back,” can you find a clean formula for its self‑convolution?
  • We already know such a formula for the Fibonacci sequence (the k=2 case). What are the matching formulas for Narayana numbers (k=3), for 4‑step versions (k=4), and for any k in general?
  • Is there a pattern in the “mysterious” constants that appear in these formulas?

How do they study it?

Here’s the idea in everyday language:

  • A recurrence is a recipe: to get a new number, combine earlier numbers in the same list. For example:
    • Fibonacci: Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2}
    • 3‑step Narayana: Rn=Rn−1+Rn−3R_n = R_{n-1} + R_{n-3}
    • k‑step Narayana: Rn=Rn−1+Rn−k\mathcal{R}_n = \mathcal{R}_{n-1} + \mathcal{R}_{n-k}
  • A self‑convolution is like “mix and match pairs that add up to the same position.” For a fixed nn, you multiply the first term with the nnth, the second term with the (n−1)(n-1)th, and so on, then add them all:

∑i=0nai an−i.\sum_{i=0}^n a_i\,a_{n-i}.

  • The main tool they use is a generating function. Think of it as turning the whole sequence into a single “function‑code” where multiplying two such codes combines sequences in a predictable way. In symbols, if

G(x)=∑n≥0Rnxn,G(x) = \sum_{n\ge 0} \mathcal{R}_n x^n,

then the self‑convolution corresponds to G(x)2G(x)^2. By writing G(x)G(x) as a simple fraction and carefully differentiating and rearranging, they match both sides of the identity. Along the way, they use standard sums (like geometric series) and a few short lemmas to keep track of coefficients.

What did they find?

  1. Known case (Fibonacci, k=2). They start from a classic result:

5∑i=0nFiFn−i=(2n)Fn+1−(n+1)Fn.5 \sum_{i=0}^n F_i F_{n-i} = (2n)F_{n+1} - (n+1)F_n.

  1. Narayana case (k=3). They confirm a similar identity for RnR_n (where Rn=Rn−1+Rn−3R_n = R_{n-1}+R_{n-3}):

31∑i=0nRiRn−i=9(n+1)Rn+2−3(n+3)Rn+1−2(n+2)Rn.31\sum_{i=0}^n R_i R_{n-i} = 9 (n + 1) R_{n + 2} - 3 (n + 3) R_{ n + 1} - 2 (n + 2) R_n.

  1. 4‑step case (k=4). They prove:

(283)∑i=0nSiSn−i=64(n+2)Sn+3−16(n+5)Sn+2−12(n+4)Sn+1−9(n+3)Sn,(283)\sum_{i=0}^n S_i S_{n-i} = 64 (n + 2) S_{n + 3} - 16 (n + 5) S_{ n + 2} - 12(n+4)S_{n+1} - 9(n+3)S_n,

where Sn=Sn−1+Sn−4S_n = S_{n-1} + S_{n-4}.

  1. A striking pattern. The big number out front (5, 31, 283, …) is not random. For k‑step Narayana numbers it is:

kk+(k−1)k−1.k^k + (k-1)^{k-1}.

For k=2, 3, 4, this gives 22+11=52^2+1^1=5, 33+22=313^3+2^2=31, 44+33=2834^4+3^3=283.

  1. The general rule (the main theorem). For the k‑step Narayana numbers Rn\mathcal{R}_n defined by
  • R0=0\mathcal{R}_0=0,
  • R1=⋯=Rk−1=1\mathcal{R}_1=\cdots=\mathcal{R}_{k-1}=1,
  • Rn=Rn−1+Rn−k\mathcal{R}_n=\mathcal{R}_{n-1}+\mathcal{R}_{n-k} for n≥kn\ge k,

the self‑convolution satisfies:

(kk+(k−1)k−1)∑i=0nRi Rn−i=k k−1(n+k−2) Rn+k−1−∑j=0k−2(kj(k−1)k−2−j) (n+k+j−1) Rn+j.(k^k + (k-1)^{k-1})\sum_{i=0}^n \mathcal{R}_i\,\mathcal{R}_{n-i} = k^{\,k-1}(n + k - 2)\,\mathcal{R}_{n + k - 1} - \sum_{j=0}^{k-2} \Big(k^j (k-1)^{k-2-j}\Big)\,(n + k + j - 1)\,\mathcal{R}_{n + j}.

In words: a weighted sum of all “pair‑products that land at position n” equals a neat weighted combination of a few nearby terms in the sequence. This one formula reproduces the Fibonacci, 3‑step, and 4‑step cases automatically, and also works for any k (like 6‑step, 10‑step, and so on).

Why is this important?

  • It reveals order in what looks like chaos. Self‑convolutions can be messy sums, but these identities turn them into simple expressions. That’s powerful for both theory and computations.
  • It unifies many sequences. Fibonacci and Lucas numbers are famous, but this shows a single pattern covers them and their Narayana‑style generalizations.
  • It connects to deep number patterns. The constants kk+(k−1)k−1k^k + (k-1)^{k-1} relate to the “shape” of the formulas behind these sequences and tie into known constants like those behind the golden and “supergolden” ratios.
  • It opens doors for future work. Understanding these identities can help in combinatorics, number theory, and even computer algorithms where such recurrences and convolutions appear (for instance, in counting problems and signal processing ideas that use similar math).

Overall, the paper turns a complicated sum into a clean rule that works for a whole family of sequences, showing that beautiful patterns extend far beyond Fibonacci.

View Paper Prompt  View All Prompts 

Knowledge Gaps

Below is a concise list of the main knowledge gaps, limitations, and open questions left by the paper. Each item is stated to be concrete and actionable for follow-up work.

  • Discriminant connection unproven in general: The paper observes that the leading multiplier matches the (signed) discriminant of 1 − x − xk for k = 2, 3, 4 and equals kk + (k−1){k−1}. A general proof and precise sign convention for all k is not provided; establishing this link via resultants/discriminants remains open.
  • Minimality/uniqueness of the multiplier: It is unclear whether kk + (k−1){k−1} is the smallest or unique integer allowing an integer-coefficient identity of the stated form. Characterize all possible integer multipliers and determine minimality.
  • Dependence on initial conditions: The result is proved only for the specific Narayana-type seeds (0, then 1’s through index k−1). Generalize the identity to arbitrary initial vectors and determine how coefficients change.
  • Generalized coefficients in the recurrence: Extend from a_n = a_{n−1} + a_{n−k} to a_n = α a_{n−1} + β a_{n−k} (α, β integers/rationals), and more broadly to sparse higher-order recurrences a_n = ∑{i∈S} c_i a{n−i}. Identify the analogue of kk + (k−1){k−1} and the right-hand-side coefficients.
  • Lucas-type companions and cross-convolutions: Develop Lucas-type companion sequences for a_n = a_{n−1} + a_{n−k} and derive (i) self-convolution formulas for these companions and (ii) cross-convolution identities between Narayana-type and Lucas-type sequences.
  • Combinatorial interpretation: Provide a direct combinatorial proof and interpretation of the coefficients kj (k−1){k−2−j} and the linear factors (n + k + j − 1), ideally in terms of known models for Narayana’s cows and their k-step analogues.
  • Asymptotics of the self-convolution: Derive explicit asymptotics of S_n = ∑{i=0}n R_i R{n−i} of the form S_n ∼ C_k n ρ_kn, where ρ_k is the dominant root of xk = x{k−1} + 1. Express C_k in terms of ρ_k and derivatives of the characteristic polynomial.
  • Recurrence for the convolution sequence itself: Identify the linear recurrence (order, coefficients) satisfied by S_n = ∑{i=0}n R_i R{n−i} and provide initial conditions; compare with the closed-form identity in the paper.
  • Higher-fold convolutions and shifted convolutions: Generalize to m-fold self-convolutions and to shifted sums ∑{i} R{i+a} R_{n−i+b}. Determine whether analogous closed forms exist and what multipliers and coefficients arise.
  • Modular/congruence phenomena: Study primes p dividing kk + (k−1){k−1} and the resulting congruences for the self-convolution modulo p. Classify congruence patterns and periodicities in n and k.
  • Binet/roots-of-unity filter approach: Provide an alternative proof using Binet-type formulas and partial fractions. Relate the identity to sums over pairs of roots and to derivatives at the dominant root; clarify the algebraic origin of the coefficients.
  • Computational validation and scalability: Supply systematic computational checks for larger k and n, along with efficient algorithms (and complexity analysis) to evaluate the convolution and verify the identity symbolically/numerically.
  • Relationship to k-bonacci/k-Lucas results: Clarify structural similarities and differences between the present “sparse” k-step recurrence (a_{n−1}+a_{n−k}) and the dense k-bonacci (sum of last k terms). Determine whether analogous convolution identities exist across these families and map any transformations between them.
  • Generating function refinements: Further analyze x2/(1 − x − xk)2 (pole structure, partial fraction decomposition) to simplify or reparameterize the right-hand-side coefficients; explore whether the identity admits a shorter or more symmetric formulation.
  • Extensions to weighted convolutions: Investigate weighted sums ∑{i} w(i) R_i R{n−i} (e.g., w(i)=i, i2, or polynomial weights) and determine whether similarly structured closed forms exist.
  • Cross-order convolutions: Explore convolution identities between sequences of different orders (e.g., k-step with ℓ-step Narayana-type sequences), including necessary multipliers and resulting coefficient structures.
  • Applications and interpretations: Identify applications in combinatorics (tilings, compositions with gaps), number theory, or dynamical models of “aging” processes; leverage the identity to count structured pairs or to derive new combinatorial statistics.
  • Formal-power-series rigor and edge cases: Clean up minor typographical and notational issues in the proof (e.g., derivative expressions and missing braces) and explicitly address edge cases k = 2, 3 in the derivation to ensure complete formal power series rigor.
View Paper Prompt  View All Prompts 

Practical Applications

Immediate Applications

Below are practical, deployable applications that leverage the paper’s main theorem (a closed-form identity for the self-convolution of k-step Narayana sequences defined by Rn=Rn−1+Rn−k\mathcal{R}_n=\mathcal{R}_{n-1}+\mathcal{R}_{n-k} with specified initial conditions), its generating-function method, and explicit examples for k=2,3,4.

  • Efficient evaluation of convolution sums in recurrence-driven computations (software, HPC)
    • Use case: Replace the naive O(n) computation of self-convolutions ∑i=0nRiRn−i\sum_{i=0}^n \mathcal{R}_i\mathcal{R}_{n-i} with the closed-form identity from Theorem 1, yielding O(k) or even O(1) time per n once neighboring Rn+j\mathcal{R}_{n+j} are available.
    • Tools/products/workflows:
    • A lightweight library function (e.g., in Python/Sage/Julia/C++) that computes k-step Narayana numbers and their self-convolutions via the theorem; unit-tested against small n.
    • Integration into dynamic programming pipelines or coefficient-computation routines for generating functions.
    • Assumptions/dependencies: Correct initial conditions (R0=0\mathcal{R}_0=0 and Ri=1\mathcal{R}_i=1 for 1≤i≤k−11\le i\le k-1), integer arithmetic or big-integer support to avoid overflow, and sequence values Rn+j\mathcal{R}_{n+j} precomputed via the recurrence.
  • Validation and QA of sequence generators (software engineering)
    • Use case: Property-based testing for code that generates k-step sequences (including Fibonacci and Narayana’s cows) by checking the convolution identity against computed values.
    • Tools/products/workflows: Automated test suites in CI pipelines; differential testing against CAS (Sage/Maple/Mathematica).
    • Assumptions/dependencies: Deterministic recurrences and reproducible numeric computations.
  • Educational modules for generating functions, Cauchy products, and linear recurrences (education)
    • Use case: Classroom or self-study materials that demonstrate deriving identities via generating functions; connecting OEIS entries to proofs.
    • Tools/products/workflows: Jupyter notebooks, interactive demos showing the identity for k=2,3,4 and general k; problem sets that guide students through reindexing, geometric sums, and term-by-term differentiation of power series.
    • Assumptions/dependencies: Background in discrete math and basic calculus; access to CAS or notebooks.
  • OEIS curation and sequence analytics (academia)
    • Use case: Documenting self-convolution identities for generalized Narayana sequences and cross-referencing known entries (e.g., Fibonacci/Lucas analogs), adding verified data and proofs.
    • Tools/products/workflows: CAS computations for sequence terms; scripts to generate tables of convolution values; submission of identity variants to the OEIS.
    • Assumptions/dependencies: Proper mapping between sequence definitions and OEIS entries; reproducible verification of examples.
  • Deterministic autocorrelation checks for linear recurrence sequences (communications/coding theory)
    • Use case: For integer sequences used in deterministic signal design or coding exercises, the identity provides a fast way to compute auto-correlations at lag n without full convolution.
    • Tools/products/workflows: Small-scale signal design labs; in-class demonstrations of convolution identities.
    • Assumptions/dependencies: Sequences are over integers; for practical coding applications over finite fields (e.g., LFSRs), modular analogues are needed (see Long-Term Applications).
  • Age-structured population or cohort interaction metrics for Narayana’s cows-like models (agriculture/ecology)
    • Use case: In models with delayed reproduction (e.g., Narayana’s cows where Rn=Rn−1+Rn−3R_n=R_{n-1}+R_{n-3}), the self-convolution can estimate pair counts or interactions stratified by “age-sum equals n.”
    • Tools/products/workflows: Spreadsheet or Python models that compute cohort interactions using the identity rather than full pairwise summations.
    • Assumptions/dependencies: The cohort/interactions genuinely correspond to the recurrence; interpretations of the convolution as pair counts depend on model design and may require careful mapping from “age-sum” to interactions.

Long-Term Applications

The following applications will likely require methodological extensions (e.g., adapting results to stochastic settings, finite fields, or different initial conditions), scaling, or further theoretical development.

  • Symbolic rewrite systems for recurrences in CAS and compilers (software)
    • Use case: A transformation engine that recognizes convolution sums of linear recurrences of the form Rn=Rn−1+Rn−k\mathcal{R}_n=\mathcal{R}_{n-1}+\mathcal{R}_{n-k} and automatically replaces them with closed-form identities to speed up computations.
    • Tools/products/workflows: CAS plugins (Mathematica/Maple/Sage), intermediate-representation passes in compilers for numeric kernels.
    • Assumptions/dependencies: Robust pattern matching, extension to broader classes of linear recurrences (e.g., with general coefficients), and correctness guarantees.
  • Autocorrelation and second-order analysis for AR(k) processes (finance/econometrics, signal processing)
    • Use case: Develop analog identities for stochastic autoregressive processes (AR(k)) that mirror the deterministic convolution identity, enabling faster computation of expected autocorrelations or moment relations.
    • Tools/products/workflows: Theoretical extensions, simulation studies, and estimation toolkits.
    • Assumptions/dependencies: Generalizing from deterministic integer recurrences to stochastic processes with noise; ensuring stationarity and proper initial distributions.
  • Cryptography and communications: modular analogues for LFSR-like sequences (coding/cryptography)
    • Use case: Adapt the convolution identity to sequences over finite fields or rings (e.g., LFSRs), providing insights into autocorrelation and cross-correlation structures useful for sequence design and analysis.
    • Tools/products/workflows: Research prototypes for modular generating functions; sequence-analysis tools for spread-spectrum or stream ciphers.
    • Assumptions/dependencies: Theory for modular generating functions and convolution identities; verification against known cryptographic sequence properties.
  • Fast cohort contact computations in epidemiological models with delayed dynamics (public health policy)
    • Use case: In age- or stage-structured epidemic models where cohort sizes follow delayed linear recurrences, use self-convolution identities to rapidly compute pairwise contact matrices or interaction counts parameterized by combined “ages.”
    • Tools/products/workflows: Modeling frameworks (e.g., compartmental models with discrete cohorts), numerical pipelines for scenario analysis.
    • Assumptions/dependencies: Realistic mapping of epidemiological dynamics to simplified recurrences; calibration with empirical data; extension to heterogeneous contact rates.
  • Optimization and control for systems with k-step memory (robotics/control)
    • Use case: When cost functions depend on pairwise combinations of state histories governed by k-step recurrences, convolution identities may reduce the complexity of evaluating these costs in trajectory planning or model predictive control.
    • Tools/products/workflows: Control software that leverages recurrence-aware cost evaluations; theoretical work to connect deterministic sequence identities to control-performance metrics.
    • Assumptions/dependencies: System dynamics must be representable or approximable by linear recurrences of the studied form; additional engineering constraints may require further adaptation.
  • Scalable sequence analytics platforms (software, data science)
    • Use case: Build interactive “sequence explorer” tools that let users define k, compute sequence terms, visualize growth, and obtain convolution-derived metrics instantly, supporting research, pedagogy, and exploratory analysis.
    • Tools/products/workflows: Web apps, APIs, and visualization dashboards; integration with OEIS and CAS backends.
    • Assumptions/dependencies: Performance and precision for large n and k; big-integer and modular arithmetic backends; usability considerations for non-expert users.

Cross-cutting assumptions and dependencies

  • Applicability hinges on the exact recurrence and initial conditions: R0=0\mathcal{R}_0=0, Ri=1\mathcal{R}_i=1 for 1≤i≤k−11\le i\le k-1, and Rn=Rn−1+Rn−k\mathcal{R}_n=\mathcal{R}_{n-1}+\mathcal{R}_{n-k} for n≥kn\ge k.
  • Numerical robustness: The coefficients in the identity involve terms like kk+(k−1)k−1k^k+(k-1)^{k-1}, which grow rapidly with k. Practical implementations need big-integer support, modular arithmetic, or scaling strategies.
  • Generalization requirement: Many sectoral applications (cryptography, stochastic modeling) need modular or probabilistic extensions of the identity; the current result is deterministic and over integers.
  • Model fidelity: Interpreting convolution sums as interaction counts (e.g., age-sum pairing) requires that the real system be meaningfully represented by the studied recurrence; otherwise, results serve as approximations or educational illustrations.
View Paper Prompt  View All Prompts 

Glossary

  • Binet's formula: A closed-form expression for terms of certain linear recurrence sequences (like Fibonacci) using characteristic roots. "use variations on Binet's formula to give self-convolution formulas for general second-order recurrence sequences"
  • Cauchy product rule: A rule stating that the product of two power series corresponds to the convolution of their coefficients. "the Cauchy product rule \cite[p.~36]{Wilf} tells us that the generating function for the self-convolution"
  • Discriminant: A polynomial invariant that indicates the nature of its roots; here, of the characteristic polynomial associated with a recurrence. "5, -31, and -283 are the discriminants of 1−x−xk1-x-x^k for k=2k=2, 3, and 4."
  • Generating function: A formal power series that encodes a sequence, enabling algebraic manipulation to derive identities. "Since the generating function for Rn\mathcal{R}_n is"
  • Golden ratio: The constant φ=(1+5)/2\varphi = (1+\sqrt{5})/2 with deep ties to Fibonacci-related sequences and identities. "we note that 5\sqrt{5} is part of the golden ratio \cite{BQ} related to the Fibonacci numbers"
  • kk-step Narayana numbers: A generalization of Narayana numbers defined by the recurrence Rn=Rn−1+Rn−k\mathcal{R}_{n} = \mathcal{R}_{n-1} + \mathcal{R}_{n-k}. "We define the kk-step Narayana numbers Rn\mathcal{R}_n as"
  • Lucas numbers: A sequence related to Fibonacci with the same recurrence but different initial conditions. "We note that the LnL_n in the last formula represents the nnth Lucas number."
  • Lucas-type sequences: Generalizations of Lucas numbers to higher-order recurrences. "generalizes equation (\ref{e.luccon}) to order-kk Lucas-type sequences"
  • Narayana numbers: A sequence defined by Rn=Rn−1+Rn−3R_{n} = R_{n-1} + R_{n-3} with specific initial values. "We now define the Narayana numbers RnR_n as"
  • Narayana sequence: The Narayana numbers viewed as a 3-step linear recurrence sequence generalizing Fibonacci. "the Narayana sequence (which is itself a generalization of the Fibonacci numbers)"
  • OEIS (On-Line Encyclopedia of Integer Sequences): A comprehensive online database of integer sequences and related facts. "Most of these can be found at \href{https://oeis.org/A001629}{A001629} on the On-Line Encyclopedia of Integer Sequences (OEIS)"
  • Order-kk recurrence sequences: Linear recurrences that depend on kk previous terms. "we will consider the order-kk recurrence sequences that are natural generalizations of the Narayana sequence"
  • Second-order recurrences: Linear recurrences determined by the previous two terms. "The Fibonacci numbers and the Lucas numbers are two specific (and famous) examples of second-order recurrences."
  • Self-convolution: The sum of products of sequence terms whose indices add to a fixed nn, i.e., ∑i=0naian−i\sum_{i=0}^n a_i a_{n-i}. "the sum on the left is called the {\em self-convolution} of the Fibonacci numbers."
  • Supergolden ratio: A constant analogous to the golden ratio but associated with certain higher-order recurrences (e.g., Narayana). "likewise 31\sqrt{31} for the ``supergolden" ratio \cite{Crilly} associated with the Narayana numbers."
  • Third-order recurrence: A linear recurrence depending on the previous three terms. "Rabinowitz's equation covers a general third-order recurrence defined as"
View Paper Prompt  View All Prompts 

Open Problems

We found no open problems mentioned in this paper.

Tweets

Sign up for free to view the 2 tweets with 135 likes about this paper.

Sign Up for Free