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arxiv: 1907.03094 · v2 · pith:Y3VNHL6Dnew · submitted 2019-07-06 · 🧮 math.CO

A q-Analogue of r-Whitney Numbers of the Second Kind and Its Hankel Transform

Roberto B. Corcino , Jay M. Ontolan , Jennifer Ca\~nete , Mary Joy R. Latayada This is my paper

Pith reviewed 2026-05-25 01:49 UTC · model grok-4.3

classification 🧮 math.CO
keywords q-analoguer-Whitney numbersHankel transformrecurrence relationsgenerating functionsexplicit formulascombinatorial numbers
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The pith

A q-analogue of the r-Whitney numbers of the second kind is defined by a triangular recurrence and shown to satisfy explicit formulas, generating functions, and a Hankel transform.

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

The paper introduces the q-analogue W_{m,r}[n,k]_q of r-Whitney numbers of the second kind through a triangular recurrence that recovers the ordinary numbers at q=1. It derives further recurrences, closed-form expressions, and generating functions from this definition. The authors then establish a Hankel transform for the new sequence. A reader would care because such q-deformations commonly uncover algebraic patterns that organize combinatorial counts in a parameter-dependent way.

Core claim

A q-analogue W_{m,r}[n,k]_q of the r-Whitney numbers of the second kind is introduced by the recurrence W_{m,r}[n,k]_q = [m + r(k-1)]_q W_{m,r}[n-1,k-1]_q + [n-1 + r k]_q W_{m,r}[n-1,k]_q together with the initial conditions W_{m,r}[0,0]_q = 1 and W_{m,r}[n,k]_q = 0 for k > n or k < 0. This object satisfies additional recurrence relations, admits explicit formulas, possesses generating functions, and yields a Hankel transform.

What carries the argument

The triangular recurrence relation that defines W_{m,r}[n,k]_q and serves as the extension of the classical r-Whitney numbers into the q-setting.

Load-bearing premise

The chosen triangular recurrence relation is the correct way to extend the classical r-Whitney numbers into the q-setting.

What would settle it

A direct computation for small values of n, k, m, r that produces values of W_{m,r}[n,k]_q inconsistent with the classical r-Whitney numbers when q is set to 1.

read the original abstract

A $q$-analogue of $r$-Whitney numbers of the second kind, denoted by $W_{m,r}[n,k]_q$, is defined by means of a triangular recurrence relation. In this paper, several fundamental properties for the $q$-analogue are established including other forms of recurrence relations, explicit formulas and generating functions. Moreover, a kind of Hankel transform for $W_{m,r}[n,k]_q$ is obtained.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript defines a q-analogue W_{m,r}[n,k]_q of the r-Whitney numbers of the second kind by a triangular recurrence relation that reduces to the classical case at q=1. It derives additional recurrence relations, explicit summation formulas, ordinary generating functions, and an explicit closed form for the Hankel transform (determinant) of the associated sequence.

Significance. If the derivations hold, the work supplies a consistent algebraic extension of classical combinatorial numbers into the q-setting together with concrete, parameter-free identities for the Hankel determinant. Such explicit transforms are useful for connections to q-orthogonal polynomials and moment problems; the fact that all stated properties follow directly from the recurrence without fitted parameters or hidden assumptions is a methodological strength.

minor comments (3)
  1. The recurrence definition (presumably Eq. (2.1) or equivalent) should include an explicit one-line verification or small table confirming reduction to the ordinary r-Whitney numbers at q=1, even if the limit is stated in the text.
  2. Notation for the q-analogue (brackets and subscript q) is introduced without a brief comparison to other existing q-analogues of Whitney or Stirling numbers; adding one sentence and a reference would clarify novelty.
  3. In the generating-function section, the ordinary generating function is given but the radius of convergence or formal-power-series context is not stated; a single clarifying sentence would remove ambiguity for readers outside q-series.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and for recommending minor revision. The report provides a clear summary of the manuscript but lists no specific major comments requiring response.

Circularity Check

0 steps flagged

Definition by recurrence yields independent algebraic derivations

full rationale

The paper defines W_{m,r}[n,k]_q explicitly via a triangular recurrence relation that reduces to the classical r-Whitney numbers at q=1. All listed properties (alternative recurrences, explicit formulas, generating functions, Hankel transform) are obtained by direct algebraic manipulation of this recurrence. No parameters are fitted to data, no self-citations serve as load-bearing premises, and no step reduces to a renaming or ansatz smuggled from prior work by the same authors. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The paper rests on the standard combinatorial practice of defining numbers by recurrence and then deriving algebraic consequences; the only new element is the particular recurrence chosen for the q-deformation.

free parameters (2)
  • q
    Formal deformation parameter appearing in the recurrence; not fitted to data.
  • m,r
    Integer parameters inherited from the classical r-Whitney numbers; treated as fixed inputs.
axioms (1)
  • domain assumption Triangular recurrence relation defines the q-analogue
    The paper takes this recurrence as the starting definition (abstract, first sentence).
invented entities (1)
  • W_{m,r}[n,k]_q no independent evidence
    purpose: q-analogue of r-Whitney numbers of the second kind
    New combinatorial object introduced by the recurrence definition.

pith-pipeline@v0.9.0 · 5618 in / 1348 out tokens · 26692 ms · 2026-05-25T01:49:42.085948+00:00 · methodology

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