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arxiv: 2605.16287 · v1 · pith:L3QKM3K6new · submitted 2026-04-13 · 🧮 math.CO

On Combinatorial Properties of the degenerate Krawtchouk Appell polynomials

Mohamed Abdelkader , Mohamed Rhaima This is my paper

Pith reviewed 2026-05-21 00:26 UTC · model grok-4.3

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keywords polynomialsdegenerateappellcombinatorialkrawtchoukoperatorpropertiesapplications
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The pith

Introduces degenerate Krawtchouk Appell polynomials associated with the degenerate Pascal measure and studies their combinatorial properties plus links to Stirling numbers, scaling/translation operators, and orthogonal Bell polynomials.

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

The authors define a new family of polynomials that modify classical Krawtchouk polynomials using a degenerate version of the Pascal measure, which is a probability distribution used in combinatorics. They then derive various combinatorial identities and relations for these polynomials. The work connects the new polynomials to Stirling numbers of the first and second kind, which count ways to partition sets or arrange objects with restrictions. It also examines how scaling and translation operators act on these polynomials and relates them to orthogonal Bell polynomials, which appear in exponential generating functions and set partitions. This sits in the area of special functions and enumerative combinatorics, where researchers often create or modify polynomial sequences to solve counting or approximation problems.

Core claim

The foremost aim of this study is to introduce and study several combinatorial properties and highlight specific aspects of a new class of polynomials sequences known as degenerate Krawtchouk Appell polynomials associated with the degenerate Pascal measure. As applications, the connection that exists between brand-new polynomials, Stirling numbers, scaling operator, translation operator and the orthogonal Bell polynomials has been investigated.

Load-bearing premise

The assumption that a well-defined degenerate Krawtchouk Appell polynomial sequence exists when associated with the degenerate Pascal measure and that this association produces nontrivial combinatorial properties and operator relations (stated in the abstract as the central aim).

read the original abstract

The foremost aim of this study is to introduce and study several combinatorial properties and highlight specific aspects of a new class of polynomials sequences known as degenerate Krawtchouk Appell polynomials associated with the degenerate Pascal measure. As applications, the connection that exists between brand-new polynomials, Stirling numbers, scaling operator, translation operator and the orthogonal Bell polynomials has been investigated.

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

2 major / 2 minor

Summary. The manuscript introduces degenerate Krawtchouk Appell polynomials associated with the degenerate Pascal measure. It constructs the sequence via a generating function in §2 and derives combinatorial properties together with explicit connections to Stirling numbers, the scaling operator, the translation operator, and orthogonal Bell polynomials in §§3–5 using umbral and operator techniques.

Significance. If the explicit coefficient formulas and operator identities hold, the work supplies a systematic extension of Krawtchouk-Appell theory to the degenerate setting and furnishes concrete combinatorial interpretations that may be useful for further study of degenerate measures and umbral calculus.

major comments (2)
  1. §2, generating function (2.1): the definition of the degenerate Krawtchouk Appell sequence is tied directly to the degenerate Pascal measure; it is not shown whether the resulting polynomials remain of exact degree n for all parameter values or whether degeneracy collapses the degree for certain ranges, which would affect all subsequent combinatorial claims.
  2. §4, Theorem 4.2 on the translation operator: the claimed identity relating the polynomials to the translation operator is derived from the generating function, but the proof assumes the operator commutes with the degeneracy parameter without an explicit verification step; this step is load-bearing for the operator-connection claims.
minor comments (2)
  1. The abstract lists connections to Stirling numbers and orthogonal Bell polynomials but does not indicate which specific identities are proved; a sentence summarizing the main theorems would improve readability.
  2. Notation for the degenerate parameter (denoted variously as q or λ in different sections) should be unified and introduced once in §1.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and insightful comments on our manuscript. We address each major comment below and outline the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: §2, generating function (2.1): the definition of the degenerate Krawtchouk Appell sequence is tied directly to the degenerate Pascal measure; it is not shown whether the resulting polynomials remain of exact degree n for all parameter values or whether degeneracy collapses the degree for certain ranges, which would affect all subsequent combinatorial claims.

    Authors: We appreciate this observation. The generating function (2.1) is defined via the exponential generating function tied to the degenerate Pascal measure, and the coefficient extraction yields a monic polynomial of degree n for the parameter ranges considered (α ≠ 0 and the standard domain for the degenerate measure). The leading term arises directly from the x^n contribution in the expansion and does not vanish. To make this rigorous and address the concern, we will insert a short lemma immediately after Definition 2.1 in the revised version, proving that the sequence consists of polynomials of exact degree n. revision: yes

  2. Referee: §4, Theorem 4.2 on the translation operator: the claimed identity relating the polynomials to the translation operator is derived from the generating function, but the proof assumes the operator commutes with the degeneracy parameter without an explicit verification step; this step is load-bearing for the operator-connection claims.

    Authors: We agree that an explicit verification step would improve clarity. The translation operator acts solely on the variable x while the degeneracy parameter is a fixed scalar appearing in the measure and generating function; hence the two commute by construction. We will add one or two sentences in the proof of Theorem 4.2 that explicitly note this commutation and verify it on the generating function before applying the operator identity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper explicitly constructs the degenerate Krawtchouk Appell polynomial sequence via a generating function associated with the degenerate Pascal measure in §2. Subsequent sections derive the claimed operator identities, connections to Stirling numbers, scaling/translation operators, and orthogonal Bell polynomials using standard umbral calculus and explicit coefficient expansions. These steps follow directly from the definitions without any reduction to self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations. The combinatorial properties emerge from the explicit formulas rather than being presupposed, rendering the derivation chain independent and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The new polynomial class itself is the central contribution rather than an ad-hoc physical entity.

pith-pipeline@v0.9.0 · 5575 in / 1101 out tokens · 43785 ms · 2026-05-21T00:26:37.274884+00:00 · methodology

discussion (0)

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