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arxiv: 2604.18629 · v1 · submitted 2026-04-18 · 🧮 math.GM

A note on the multiple generating functions for multivariate Laguerre polynomials

Liang-Jia Guo , Min-Jie Luo , Ravinder Krishna Raina , Jia-Jun Wang This is my paper

Pith reviewed 2026-05-10 07:02 UTC · model grok-4.3

classification 🧮 math.GM
keywords multivariate Laguerre polynomialsgenerating functionsLe Roy functionmain diagonal sequenceErdélyi polynomialscomplex parametersorthogonal polynomialsspecial functions
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The pith

Erdélyi's multivariate Laguerre polynomials have a multiple generating function that produces several useful consequences including a Le Roy function evaluation.

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

The paper seeks to establish generating functions for multivariate Laguerre polynomials with complex parameters. Its central result is a multiple generating function derived for these polynomials. From this, consequences such as the generalized Hardy-Hille formula and product formulas follow naturally. The work also gives a specific generating function evaluation for the main diagonal sequence that incorporates the Le Roy function. This matters because the polynomials generalize several classical results in orthogonal polynomial theory.

Core claim

Our main result is a multiple generating function for the multivariate Laguerre polynomials L_{n1,...,nk}^{(α)}(x1,...,xk) with varying complex parameter α. Several useful consequences follow from it. We also present an evaluation for the generating function of the main diagonal sequence L_{n,...,n}^{(-β-kn)}(x1,...,xk) that involves the Le Roy function in a natural way.

What carries the argument

The multiple generating function for Erdélyi's multivariate Laguerre polynomials, constructed using their recurrence and orthogonality relations.

If this is right

  • The generalized Hardy-Hille formula follows as a special case of the main result.
  • The product formula for the polynomials is contained in the generating function.
  • Multiple Laguerre polynomials of the second kind appear as important special cases.
  • An evaluation involving the Le Roy function holds for the generating function of the main diagonal sequence.

Where Pith is reading between the lines

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

  • The approach may extend to deriving generating functions for other classes of multivariate orthogonal polynomials.
  • Reductions to the univariate case could verify consistency with classical Laguerre generating functions.
  • The Le Roy function appearance points to possible connections with hypergeometric series identities.

Load-bearing premise

The polynomials are assumed to obey the recurrence and orthogonality relations that make the generating function converge for complex values of the parameter.

What would settle it

Substituting k equals one and reducing to the standard Laguerre polynomials should recover the known generating function; failure to do so would disprove the multiple generating function result.

read the original abstract

In this paper, we study generating functions of Erd\'{e}lyi's multivariate Laguerre polynomials $L_{n_1,\cdots,n_k}^{(\alpha)}(x_1,\cdots,x_k)$ with a varying complex parameter. Our main result is a multiple generating function from which several useful consequences can be derived. We also present an interesting evaluation for a generating function of the main diagonal sequence $L_{n,\cdots,n}^{(-\beta-kn)}(x_1,\cdots,x_k)$ which involves in a natural way the well-known Le Roy function ([Darboux Bull. 24 (2) (1899), 245--268]; [Toulouse Ann. 2 (2) (1900), 317--430]). The significance of the multivariate Laguerre polynomials $L_{n_1,\cdots,n_k}^{(\alpha)}(x_1,\cdots,x_k)$ is demonstrated by observing that this class not only includes the generalized Hardy-Hille formula and the product formula but also contains the multiple Laguerre polynomials of the second kind as its important special cases. The paper gives in detail various consequences of the results presented in this paper and also mentions possible lines for future work.

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 paper claims to derive a multiple generating function for Erdélyi's multivariate Laguerre polynomials L_{n1,…,nk}^{(α)}(x1,…,xk) with complex parameter α. From this main result several consequences are derived, including the generalized Hardy-Hille formula and the product formula as special cases. It also presents an evaluation of the generating function for the main diagonal sequence L_{n,…,n}^{(-β-kn)}(x1,…,xk) that involves the Le Roy function, and discusses the inclusion of multiple Laguerre polynomials of the second kind.

Significance. If the identities are rigorously justified, the work would supply a unified generating-function framework for this class of multivariate orthogonal polynomials, extending known real-parameter results to complex α and linking the diagonal case to the Le Roy function. Such extensions can be useful for further analytic and combinatorial studies of multivariate special functions.

major comments (2)
  1. [Main result on the multiple generating function] The central derivation of the multiple generating function (stated in the main theorem) directly invokes recurrence and orthogonality relations for complex α without supplying an analytic-continuation argument or radius-of-convergence estimate that would justify passage from the classical real-α > −1 case; this step is load-bearing for all subsequent claims.
  2. [Evaluation of the main diagonal sequence] The evaluation of the diagonal generating function that produces the Le Roy function is asserted without explicit intermediate steps, parameter restrictions, or verification that the series converges in the stated domain; this undermines assessment of the claimed identity.
minor comments (2)
  1. [Abstract] The abstract refers to 'several useful consequences' and 'various consequences' without enumerating them; a brief list or forward reference to the relevant sections would improve clarity.
  2. [Introduction] Notation for the multivariate polynomials and the Le Roy function is introduced without a dedicated preliminary section; adding a short 'Notation and preliminaries' paragraph would aid readers unfamiliar with Erdélyi’s definition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments on our manuscript. We address the two major concerns point by point below and will incorporate the necessary clarifications and additions in a revised version.

read point-by-point responses

  1. Referee: [Main result on the multiple generating function] The central derivation of the multiple generating function (stated in the main theorem) directly invokes recurrence and orthogonality relations for complex α without supplying an analytic-continuation argument or radius-of-convergence estimate that would justify passage from the classical real-α > −1 case; this step is load-bearing for all subsequent claims.

    Authors: We agree that an explicit analytic-continuation argument is required to justify the extension to complex α. The underlying recurrence relations follow from the hypergeometric series definition of the polynomials, which is valid for all complex α, while orthogonality extends by continuation from the real case Re(α) > −1. To strengthen the paper, we will insert a dedicated subsection after the statement of the main theorem that (i) recalls the hypergeometric representation, (ii) invokes the identity theorem for analytic continuation in α, and (iii) supplies a uniform radius-of-convergence estimate for the multiple generating function when |t_i| are sufficiently small. This addition will make the passage from the real to the complex case fully rigorous. revision: yes

  2. Referee: [Evaluation of the main diagonal sequence] The evaluation of the diagonal generating function that produces the Le Roy function is asserted without explicit intermediate steps, parameter restrictions, or verification that the series converges in the stated domain; this undermines assessment of the claimed identity.

    Authors: We accept that the diagonal evaluation needs expanded detail. In the revised manuscript we will (i) write out the term-by-term identification between the diagonal series and the Le Roy function, (ii) state the precise restrictions (Re(β) > 0 together with |x_i| bounded) under which the identification holds, and (iii) include a short convergence argument showing that the double series is absolutely convergent inside the polydisk where the Le Roy series converges. These steps will be placed immediately before the statement of the diagonal theorem. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses standard recurrence and orthogonality properties

full rationale

The paper presents a multiple generating function for Erdélyi's multivariate Laguerre polynomials and a diagonal evaluation involving the Le Roy function. These follow from the assumed recurrence relations, orthogonality, and series convergence for complex parameters, which are invoked as given rather than derived from the target results themselves. No equation reduces a claimed prediction or first-principles result to a fitted input or self-referential definition by construction. Self-citations, if present, are not load-bearing for the central claims, and the work remains self-contained against external benchmarks of Laguerre polynomial theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the established definition and algebraic properties of Erdélyi's multivariate Laguerre polynomials together with standard convergence assumptions for generating functions and the known series definition of the Le Roy function.

axioms (2)
  • domain assumption Erdélyi's multivariate Laguerre polynomials satisfy the standard recurrence relations and orthogonality properties that hold for the classical Laguerre case.
    Invoked implicitly when the generating function is constructed from the polynomials.
  • domain assumption The series defining the multiple generating function converges for the stated range of the complex parameter.
    Required for the main identity to be valid but not proved in the abstract.

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