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arxiv: 2606.02405 · v1 · pith:MTMA3AZKnew · submitted 2026-06-01 · 🧮 math.CA

Mehler formula for Wronskians of Hermite polynomials

Alexey Kuznetsov , Minjian Yuan This is my paper

Pith reviewed 2026-06-28 11:31 UTC · model grok-4.3

classification 🧮 math.CA
keywords Hermite polynomialsWronskiansMehler formulabilinear generating functionorthogonal polynomialsexceptional Hermite polynomials
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The pith

The bilinear generating function for Wronskians of Hermite polynomials equals the classical Mehler kernel multiplied by a polynomial.

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

This paper proves that the bilinear generating function built from Wronskians of Hermite polynomials takes the form of the classical Mehler kernel multiplied by a polynomial. The result extends an earlier formula that was known only for the special case of exceptional Hermite polynomials. The authors derive several properties of the multiplier polynomials and state four conjectures about them. A reader would care because the closed form supplies an explicit expression for a wider family of objects that appear in the analysis of orthogonal polynomials and their associated differential equations.

Core claim

The bilinear generating function for Wronskians of Hermite polynomials can be expressed as the classical Mehler kernel multiplied by a polynomial. This identity holds in the general case and extends the corresponding formula previously established for exceptional Hermite polynomials. Several properties of the appearing polynomials are established, and four conjectures about them are presented.

What carries the argument

The extended Mehler formula, in which the classical kernel is multiplied by a polynomial that encodes the Wronskian structure.

If this is right

  • The multiplier polynomials satisfy a collection of explicit algebraic and differential properties.
  • Four specific conjectures about the multiplier polynomials are formulated and left open.
  • The formula supplies a closed-form expression for generating functions that involve general Wronskians rather than only exceptional ones.
  • The result recovers the classical Mehler formula when the Wronskian is taken with respect to a single Hermite polynomial.

Where Pith is reading between the lines

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

  • If the conjectures hold, the multiplier polynomials may admit a recursive construction or an explicit product formula.
  • The same pattern of kernel times polynomial may apply to Wronskians of other classical orthogonal polynomial families.
  • The identity could simplify the derivation of addition formulas or Christoffel-Darboux-type relations that involve Wronskians.

Load-bearing premise

The extension from the exceptional Hermite case to arbitrary Wronskians of Hermite polynomials holds without further restrictions on the indices or the bilinear form.

What would settle it

Explicit computation of the bilinear generating function for a small collection of distinct indices, followed by direct comparison with the Mehler kernel times a candidate polynomial factor.

read the original abstract

We prove that the bilinear generating function for Wronskians of Hermite polynomials can be expressed as the classical Mehler kernel multiplied by a polynomial, thereby extending the result of Pupasov-Maksimov for exceptional Hermite polynomials. We establish several properties of the polynomials appearing in this extended version of the Mehler formula and present four conjectures about them.

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 proves that the bilinear generating function for Wronskians of Hermite polynomials equals the classical Mehler kernel multiplied by a polynomial, extending the Pupasov-Maksimov result from the exceptional case. The multiplier is defined via an explicit determinant formula; its polynomial character is established by direct verification of degree and leading-term behavior, and the generating-function identity is derived without additional index restrictions beyond the Wronskian definition itself. Several properties of the multiplier polynomials are established and four conjectures are stated.

Significance. If the central identity holds, the result supplies a closed-form expression for a broad class of bilinear generating functions involving Hermite Wronskians. The explicit determinant construction and the direct (non-circular) verification of the polynomial property constitute a clear technical advance over the exceptional-Hermite case and may facilitate further work on orthogonal-polynomial identities and associated special-function kernels.

minor comments (2)
  1. A brief statement in the introduction or abstract indicating the content of the four conjectures would assist readers in assessing the scope of the open questions.
  2. An explicit low-order example (e.g., the 2 imes2 Wronskian case) illustrating the determinant formula and the resulting polynomial multiplier would improve readability without lengthening the paper substantially.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. We are pleased that the central identity and its extensions are viewed as a technical advance.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit construction

full rationale

The manuscript derives the bilinear generating function identity by defining the multiplier polynomial explicitly via a determinant formula, then proving it is a polynomial through direct verification of degree and leading coefficients. This construction extends the Pupasov-Maksimov result (different authors) using only standard Hermite polynomial properties and Wronskian definitions, without any fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations. The central claim rests on verifiable algebraic identities rather than reducing to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, ad-hoc axioms, or invented entities; the result is framed as a proof resting on classical properties of Hermite polynomials and the Mehler kernel.

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