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

Recognition: unknown

A note on the multiple generating functions for multivariate Laguerre polynomials

Jia-Jun Wang, Liang-Jia Guo, Min-Jie Luo, Ravinder Krishna Raina

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.

Desk Editor's Note private letter to a colleague

The paper derives a multiple generating function for Erdélyi's multivariate Laguerre polynomials with complex α and a Le Roy evaluation on the diagonal, but the extension from real to complex parameters lacks explicit justification.

read the letter

The main thing here is a multiple generating function for Erdélyi's multivariate Laguerre polynomials that allows a complex parameter, plus an evaluation of the main diagonal sequence that brings in the Le Roy function in a natural way. They also recover the generalized Hardy-Hille formula and product formula as special cases and note that multiple Laguerre polynomials of the second kind sit inside the same framework. That placement in the literature is useful and keeps the claims grounded rather than overstated. The derivations appear to rest on the standard recurrences and orthogonality relations for these polynomials. The soft spot is the complex parameter. Those relations are classically stated for real α greater than -1, and the abstract gives no separate convergence estimate or analytic continuation step to justify using them when α is complex. If the full text does not supply that argument, the main identities are only secure in the real case. This is narrow work aimed at people already working on multivariate orthogonal polynomials and their generating functions. A reader who needs exactly these identities might find them handy for further derivations, but the paper does not open new directions or reshape anything larger. It shows honest engagement with the existing results on these polynomials and is coherent enough on its own terms to deserve referee time. I would send it to peer review with a request that the referees check the justification for complex α.

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)

[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.
[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)

[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.
[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

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
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.

pith-pipeline@v0.9.0 · 5521 in / 1379 out tokens · 48777 ms · 2026-05-10T07:02:15.533113+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 2 canonical work pages

[1]

Abdul-Halim, W .A

N. Abdul-Halim, W .A. Al-Salam, Double Euler transformations of certain hypergeometric functions, Duke Math. J. 30 (1963), 51–62

1963
[2]

Akta¸ s, E

R. Akta¸ s, E. Erku¸ s-Duman, The Laguerre polynomials in several variables, Math. Slovaca 63 (2) (2013), 531–544

2013
[3]

Ben Cheikh, I

Y. Ben Cheikh, I. Lamiri, Generating functions via integral transforms, J. Math. Anal. Appl. 331 (2) (2007), 1200– 1229

2007
[4]

Bleher, A.B.J

P .M. Bleher, A.B.J. Kuijlaars, Integral representations for multiple Hermite and multiple Laguerre polynomials, Ann. Inst. Fourier 55 (6) (2005), 2001–2014

2005
[5]

Carlitz, An integral for the product of two Laguerre polynomials, Boll

L. Carlitz, An integral for the product of two Laguerre polynomials, Boll. Unione Mat. Ital., III. Ser. 17 (1962), 25–28. 16 L.-J. Guo, M.-J. Luo, R.K. Raina and J.-J. Wang

1962
[6]

Carlitz, Bilinear generating functions for Laguerre and Lauricella polynomials in several variables, Rend

L. Carlitz, Bilinear generating functions for Laguerre and Lauricella polynomials in several variables, Rend. Sem. Mat. Univ. Padova 43, (1970) 269–276

1970
[7]

Carlitz, H.M

L. Carlitz, H.M. Srivastava, Some hypergeometric polynomials associated with the Lauricella functionF D of sev- eral variables. I, Mat. Vesn., N. Ser. 13 (28) (1976), 41–47

1976
[8]

Carlitz, H.M

L. Carlitz, H.M. Srivastava, Some hypergeometric polynomials associated with the Lauricella functionF D of sev- eral variables. II, Mat. Vesn., N. Ser. 13 (28) (1976), 143–152

1976
[9]

Chak, Some generalizations of Laguerre polynomials-II, Mat

A.M. Chak, Some generalizations of Laguerre polynomials-II, Mat. Vesnik (7) (22) (1970), 14–18

1970
[10]

Demenin, On a class of orthogonal polynomials in several variables, Ukr

A.N. Demenin, On a class of orthogonal polynomials in several variables, Ukr. Math. J. 23 (1971), 324–328

1971
[11]

Erdélyi, Beitrag zur Theorie der konfluenten hypergeometrischen Funktionen von mehreren Veränderlichen, Sitzungsber

A. Erdélyi, Beitrag zur Theorie der konfluenten hypergeometrischen Funktionen von mehreren Veränderlichen, Sitzungsber. Akad. Wiss. Wien, Math.-Naturw. Kl., Abt. IIa 146 (1937), 431–467

1937
[12]

Exton, Two new multivariable generating relations, Ark

H. Exton, Two new multivariable generating relations, Ark. Mat. 30 (2) (1992), 245–258

1992
[13]

Feldheim, Contributi alla teoria delle funzioni ipergeometriche di più variabili, Ann

E. Feldheim, Contributi alla teoria delle funzioni ipergeometriche di più variabili, Ann. Sc. Norm. Super. Pisa, II. Ser. 12 (1943), 17–59

1943
[14]

Garrappa, S

R. Garrappa, S. Rogosin, F . Mainardi, On a generalized three-parameter Wright function of Le Roy type, Fract. Calc. Appl. Anal. 20 (5) (2017), 1196–1215

2017
[15]

Gerhold, Asymptotics for a variant of the Mittag-Leffler function, Integral Transforms Spec

S. Gerhold, Asymptotics for a variant of the Mittag-Leffler function, Integral Transforms Spec. Funct. 23 (6) (2012), 397–403

2012
[16]

P .-C. Hang, L. Hu, M.-J. Luo, Asymptotic expansions of the Humbert functionΦ 1 and their applications, 2026. Available online athttps://arxiv.org/abs/2504.09280v3

work page arXiv 2026
[17]

Le Roy, Valeurs asymptotiques de certaines séries procédant suivant les puissances entières et positives d’unevariable réelle (French), Darboux Bull

É. Le Roy, Valeurs asymptotiques de certaines séries procédant suivant les puissances entières et positives d’unevariable réelle (French), Darboux Bull. 2 (24) (1899), 245–268
[18]

Le Roy, Sur les séries divergentes et les fonctions définies par un développement de Taylor

É. Le Roy, Sur les séries divergentes et les fonctions définies par un développement de Taylor. Toulouse Ann. 2 (2) (1900), 317–430

1900
[19]

Lee, Properties of multiple Hermite and multiple Laguerre polynomials by the generating function, Integral Transforms Spec

D.W . Lee, Properties of multiple Hermite and multiple Laguerre polynomials by the generating function, Integral Transforms Spec. Funct. 18 (11-12) (2007), 855–869

2007
[20]

Lee, Structure relations of classical multiple orthogonal polynomials by a generating function, J

D.W . Lee, Structure relations of classical multiple orthogonal polynomials by a generating function, J. Korean Math. Soc. 50 (5) (2013), 1067–1082

2013
[21]

Liu, S.-D

S.-J. Liu, S.-D. Lin, H.-C. Lu, H.M. Srivastava, Linearization of the products of the generalized Lauricella polyno- mials and the multivariate Laguerre polynomials, Stud. Sci. Math. Hung. 50 (3) (2013), 373–391

2013
[22]

Liu, S.-D

S.-J. Liu, S.-D. Lin, H.-C. Lu, H.M. Srivastava, Linearization of the products of the Carlitz-Srivastava polynomials of the first and second kinds via their integral representations, Appl. Math. Comput. 219 (9) (2013), 4545–4550

2013
[23]

Luo, R.K

M.-J. Luo, R.K. Raina, Upper bounds for Erdélyi’ s multivariate Laguerre polynomials, accepted byJournal of Math- ematical Inequalities, 2025. Available online athttps://arxiv.org/abs/2507.11024

work page arXiv 2025
[24]

Melczer, An Invitation to Analytic Combinatorics: From One to Several Variables

S. Melczer, An Invitation to Analytic Combinatorics: From One to Several Variables. Texts & Monographs in Sym- bolic Computation. Cham: Springer, 2021

2021
[25]

McBride, Obtaining Generating functions, Springer, Berlin (1971)

E.B. McBride, Obtaining Generating functions, Springer, Berlin (1971)

1971
[26]

Olver, D.W

F .W .J. Olver, D.W . Lozier, R.F . Boisvert, C.W . Clark (Eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, 2010

2010
[27]

Özarslan, On a singular integral equation including a set of multivariate polynomials suggested by Laguerre polynomials, Appl

M.A. Özarslan, On a singular integral equation including a set of multivariate polynomials suggested by Laguerre polynomials, Appl. Math. Comput. 229 (2014), 350–358

2014
[28]

Özmen, Some new generating functions for the modified Laguerre polynomials, Adv

N. Özmen, Some new generating functions for the modified Laguerre polynomials, Adv. Appl. Math. Mech. 11 (6) (2019), 1398–1414

2019
[29]

Pemantle, M.C

R. Pemantle, M.C. Wilson, Analytic Combinatorics in Several Variables, Cambridge Studies in Advanced Mathe- matics, vol. 140, Cambridge University Press, Cambridge, 2013

2013
[30]

Rainville, Special Functions, The Macmillan Company, New York, 1960

E.D. Rainville, Special Functions, The Macmillan Company, New York, 1960

1960
[31]

Rogosin, M

S. Rogosin, M. Dubatovskaya, Multi-parametric Le Roy function, Fract. Calc. Appl. Anal. 26 (1) (2023), 54–69

2023
[32]

Sánchez-Ruiz, P

J. Sánchez-Ruiz, P . López-Artés, J.S. Dehesa, Expansions in series of varying Laguerre polynomials and some ap- plications to molecular potentials, J. Comput. Appl. Math. 153 (1-2) (2003), 411–421

2003
[33]

Srivastava, Some multilinear generating functions, Rend

H.M. Srivastava, Some multilinear generating functions, Rend. Circ. Mat. Palermo, II. Ser. 33 (1984), 5–33

1984
[34]

Srivastava, P .W

H.M. Srivastava, P .W . Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood Ltd., Chichester, 1985. A note on the multiple generating functions for multivariate Laguerre polynomials 17

1985
[35]

Srivastava, H.L

H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chich- ester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1984

1984
[36]

Srivastava, J.P

H.M. Srivastava, J.P . Singhal, Some formulas involving the products of several Jacobi or Laguerre polynomials, Acad. roy. Belgique, Bull. Cl. Sci., V . Ser. 58 (1972), 1238–1247

1972
[37]

Tremblay, M.L

R. Tremblay, M.L. Lavertu, P . Humbert’ s confluent hypergeometric functionφ 1(α,β,γ;x,y), Jñ ¯an¯abha, Sect. A 2 (1972), 11–18

1972
[38]

Zhang, G

L. Zhang, G. Filipuk, On certain Wronskians of multiple orthogonal polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 103, 19p

2014