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arxiv: 2512.21072 · v2 · submitted 2025-12-24 · 🧮 math.CO

The Multivariable Generalized Hermite-Type-Genocchi Polynomials of Order a

Roberto B. Corcino , Cristina B. Corcino This is my paper

Pith reviewed 2026-05-16 20:16 UTC · model grok-4.3

classification 🧮 math.CO
keywords multivariable polynomialsGenocchi polynomialsHermite-type polynomialsgeneralized Stirling numbersaddition formulasexplicit representationspolynomial expansions
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The pith

A new family of multivariable generalized Hermite-type-Genocchi polynomials of order a is constructed by integration and shown to satisfy explicit representations, addition formulas, polynomial expansions, and relations to generalized Stir­

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

The paper constructs a new class of polynomials by integrating existing polynomials to create the multivariable generalized Hermite-type-Genocchi polynomials of order a. It then investigates this family by deriving explicit representations for the polynomials, addition formulas that combine them, and expansions expressing other polynomials in this basis. Relationships are also established with generalized Stirling numbers of the first kind. A sympathetic reader would care because this extends the toolkit of special polynomials used in combinatorics and analysis, allowing for new ways to handle multivariable problems and identities.

Core claim

The multivariable generalized Hermite-type-Genocchi polynomials of order a are introduced as a new class obtained through integration. They are shown to have explicit representations, to obey addition formulas, to serve in polynomial expansions, and to be connected to generalized Stirling numbers of the first kind.

What carries the argument

The integration process used to define the multivariable generalized Hermite-type-Genocchi polynomials of order a from base polynomials, which then supports deriving all the listed properties.

If this is right

  • Explicit representations give a direct way to write out the polynomials.
  • Addition formulas enable combining polynomials with shifted arguments or parameters.
  • Polynomial expansions allow any polynomial to be written as a linear combination of these new ones.
  • Links to generalized Stirling numbers of the first kind provide combinatorial meaning and additional identities.

Where Pith is reading between the lines

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

  • The integration method might be applicable to create similar extensions for other polynomial families like Euler or Bernoulli polynomials.
  • These new relations could lead to generalizations of known identities involving Stirling numbers in a multivariable setting.
  • Such polynomials may find use in generating functions for counting problems with multiple parameters.

Load-bearing premise

The integration process produces a consistent and well-defined family of polynomials for which the explicit representations, addition formulas, expansions, and Stirling relations hold without contradictions for arbitrary order a and number of variables.

What would settle it

A calculation of the integrated definition for low order a, such as a equals 1 with one or two variables, that does not match the provided explicit form or the claimed connection to the generalized Stirling numbers of the first kind.

read the original abstract

This study presents a new class of poly-Genocchi polynomials constructed through the integration of some interesting polynomials. The resulting family, referred to as the multivariable generalized Hermite-type-Genocchi polynomials of order a, is investigated in detail. Several fundamental properties are derived, including explicit representations, addition formulas, and polynomial expansions. In addition, relationships between this new family of polynomials and a certain generalized Stirling numbers of the first kind are established.

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 / 2 minor

Summary. The manuscript defines a new family of multivariable generalized Hermite-type-Genocchi polynomials of order a via an integration operator applied to a base family of polynomials. It derives explicit representations, addition formulas, polynomial expansions, and relations to a certain class of generalized Stirling numbers of the first kind.

Significance. If the derivations hold, the work adds a new parameterized multivariable extension to the literature on Genocchi-type polynomials. The explicit links to generalized Stirling numbers of the first kind supply combinatorial interpretations that may prove useful for enumeration problems. The constructive definition via integration is a standard technique whose consequences are typically straightforward once the generating function is fixed.

minor comments (2)
  1. [Definition] The definition section should state the precise generating function or integral kernel used for the base Hermite and Genocchi polynomials so that the multivariable extension is unambiguously specified.
  2. [Explicit representations] In the explicit representation, confirm that the formula is consistent when the number of variables is reduced to one and that the order parameter a appears correctly in all terms.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our work on the multivariable generalized Hermite-type-Genocchi polynomials of order a. The recommendation for minor revision is noted. No specific major comments were raised in the report, so we have no individual points to address. We will perform a final proofreading pass to ensure the manuscript is free of any minor typographical or formatting issues before resubmission.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces the multivariable generalized Hermite-type-Genocchi polynomials of order a via a constructive integral operator applied to a base family of polynomials. All listed properties—explicit representations, addition formulas, polynomial expansions, and relations to generalized Stirling numbers of the first kind—are derived as direct algebraic consequences of this definition and the associated generating function. No equation reduces a claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation chain; the derivations remain independent once the integral construction is fixed. This is a standard constructive approach in special-functions literature with no internal reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work rests on standard definitions of Hermite and Genocchi polynomials plus the assumption that their integration yields a new valid family; no fitted numerical parameters are mentioned.

axioms (1)
  • standard math Standard algebraic properties of polynomials and generating functions hold for the base Hermite and Genocchi families
    Invoked implicitly when constructing the new family via integration.
invented entities (1)
  • multivariable generalized Hermite-type-Genocchi polynomials of order a no independent evidence
    purpose: New polynomial family unifying and extending Hermite and Genocchi types
    Introduced in the paper as the central object; no independent external evidence or falsifiable prediction is provided in the abstract.

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Reference graph

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