On the Macdonald-type function and its relation with index transforms and orthogonal polynomials

Semyon Yakubovich

arxiv: 2606.05329 · v1 · pith:AWJB77MPnew · submitted 2026-06-03 · 🧮 math.CA

On the Macdonald-type function and its relation with index transforms and orthogonal polynomials

Semyon Yakubovich This is my paper

Pith reviewed 2026-06-28 03:26 UTC · model grok-4.3

classification 🧮 math.CA

keywords Macdonald functionrecurrence relationsassociated Laguerre polynomialsmultiple orthogonal polynomialsindex transformsspecial functions

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The pith

The Macdonald-type function M_ν(z) obeys recurrence relations derived from associated Laguerre polynomials and supports multiple orthogonal polynomials on the scaled weights 2 x^{ν/2} M_ν(2√x).

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

The paper continues the investigation of M_ν(z), the function tied to the Macdonald function K_ν(z) by a Fourier integral. Recurrence relations are derived for M_ν(z) and its derivatives that make direct use of properties of associated Laguerre polynomials. Multiple orthogonal polynomials are then examined for the weights hô_ν(x) = 2 x^{ν/2} M_ν(2√x) with x > 0. A sympathetic reader would care because the relations give a concrete way to compute or manipulate the function through known polynomials and because the weights yield new families of orthogonal polynomials linked to the Macdonald kernel.

Core claim

Recurrence relations for the Macdonald-type function M_ν(z) and its derivatives are obtained that involve the associated Laguerre polynomials; these relations in turn allow the construction and study of multiple orthogonal polynomials associated with the scaled Macdonald-type weights hô_ν(x) = 2 x^{ν/2} M_ν(2√x) for x > 0.

What carries the argument

The Macdonald-type function M_ν(z) defined by the Fourier integral with K_ν(z), which supplies both the recurrence relations and the weight functions for the orthogonal polynomials.

If this is right

Recurrence relations hold for M_ν(z) and its derivatives using properties of associated Laguerre polynomials.
Multiple orthogonal polynomials can be defined and studied for the scaled weights hô_ν(x) = 2 x^{ν/2} M_ν(2√x).
The relations provide a route to analyze or compute M_ν(z) by appealing to the known theory of Laguerre polynomials.

Where Pith is reading between the lines

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

The same recurrences may yield explicit series or generating-function representations for M_ν(z) itself.
The resulting orthogonal polynomials could furnish new integral transforms or index transforms that extend those already associated with K_ν(z).

Load-bearing premise

The function M_ν(z) is well-defined by its Fourier integral association with K_ν(z) and possesses sufficient analytic properties to support the stated recurrence relations and the construction of the associated multiple orthogonal polynomials.

What would settle it

Direct numerical substitution of specific values of ν and z into the claimed recurrence relations yields a mismatch, or the polynomials constructed from the weight hô_ν(x) fail to satisfy the expected multiple orthogonality conditions.

read the original abstract

We continue to investigate properties of the function $M_\nu(z)$ which is associated with the Macdonald function $K_\nu(z)$ in terms of the corresponding Fourier integral. In particular, recurrence relations for this function and its derivatives are obtained, involving properties of the associated Laguerre polynomials. Multiple orthogonal polynomials related to the scaled Macdonald-type weights $ \hat{\rho}_{\nu}(x)= 2 x^{\nu/2} M_\nu\left(2\sqrt x\right), x >0$ are 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.

Desk Editor's Note private letter to a colleague

Follow-up paper deriving recurrences for M_ν(z) and studying related multiple OPs, but weight positivity and convergence rest on unshown details.

read the letter

The one or two things to know: this is a follow-up paper on the Macdonald-type function M_ν(z) that gets recurrence relations with Laguerre polynomials and looks at multiple orthogonal polynomials with a scaled version of it as weight.

The paper does the job of extending the previous investigations in a straightforward manner. The recurrences are the main new result, and tying them to the polynomials is a natural step in this area of special functions.

The soft spot is the foundation for the weight function. The definition via Fourier integral with K_ν(z) is given, but there is no discussion in the abstract of convergence conditions or positivity for x > 0 after scaling. If M_ν(z) is not positive or the integral doesn't behave well for the ν of interest, then the multiple OP part rests on shaky ground. The stress-test note about this is on point based on the abstract alone.

The math looks like standard manipulations in the field, with no obvious circularity.

This paper is for people who work on special functions, index transforms, and orthogonal polynomials. It will be of interest mainly to those already familiar with the author's earlier papers on M_ν(z).

It deserves a serious referee to go through the derivations and confirm the properties of the weight.

Recommendation: Send it out for peer review, with the referee asked to pay attention to the analytic details of the weight.

Referee Report

1 major / 1 minor

Summary. The manuscript continues the investigation of the Macdonald-type function M_ν(z) associated with the Macdonald function K_ν(z) via a Fourier integral. It derives recurrence relations for M_ν(z) and its derivatives that involve properties of associated Laguerre polynomials. It further studies multiple orthogonal polynomials constructed from the scaled Macdonald-type weights ρ̂_ν(x) = 2 x^{ν/2} M_ν(2√x) for x > 0.

Significance. If the analytic properties of M_ν(z) and the positivity/integrability of the weight are rigorously established, the results would supply explicit recurrences linking a Macdonald-type function to Laguerre polynomials and furnish new families of multiple orthogonal polynomials tied to index transforms. Such connections are potentially useful in approximation theory and special-function identities.

major comments (1)

[Definition of M_ν(z) and weight construction] The Fourier-integral definition of M_ν(z) must be accompanied by an explicit statement of the range of ν for which the integral converges absolutely and the scaled weight ρ̂_ν(x) remains positive with all moments finite on (0,∞). This verification is load-bearing for both the recurrence derivations and the multiple-orthogonal-polynomial construction; without it the claims cannot be evaluated.

minor comments (1)

Notation for the scaled weight should be introduced with a clear reference to the original Macdonald function K_ν(z) to avoid ambiguity in the index-transform context.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comment. We address it point by point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Definition of M_ν(z) and weight construction] The Fourier-integral definition of M_ν(z) must be accompanied by an explicit statement of the range of ν for which the integral converges absolutely and the scaled weight ρ̂_ν(x) remains positive with all moments finite on (0,∞). This verification is load-bearing for both the recurrence derivations and the multiple-orthogonal-polynomial construction; without it the claims cannot be evaluated.

Authors: We agree that the manuscript currently lacks an explicit statement of the admissible range of ν. In the revised version we will add, immediately after the Fourier-integral definition of M_ν(z), a dedicated paragraph that (i) states the precise conditions on ν (derived from the known decay and analytic properties of K_ν) under which the integral converges absolutely, (ii) verifies that ρ̂_ν(x) is positive for x>0, and (iii) confirms that all moments ∫_0^∞ x^k ρ̂_ν(x) dx remain finite. These additions will directly support the subsequent recurrence relations and the multiple-orthogonal-polynomial construction. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rest on external standard functions and known polynomial properties

full rationale

The paper defines M_ν(z) via its Fourier-integral association with the independently defined Macdonald function K_ν(z), then derives recurrence relations by invoking standard properties of associated Laguerre polynomials. Multiple orthogonal polynomials are constructed from the resulting weight function. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology; all load-bearing inputs (K_ν, Laguerre) are external to the present work and not redefined in terms of the outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard analytic properties of the Macdonald function K_ν(z) and associated Laguerre polynomials, plus the Fourier integral definition of M_ν(z). No free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption The Macdonald function K_ν(z) has known Fourier integral representations and analytic properties sufficient to define M_ν(z) and derive recurrences.
Invoked by the association of M_ν(z) with K_ν(z) via Fourier integral and the use of Laguerre polynomial properties.

pith-pipeline@v0.9.1-grok · 5603 in / 1186 out tokens · 43966 ms · 2026-06-28T03:26:53.855767+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references

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11.S. Yakubovich, Orthogonal polynomials with ultra-exponential weight functions: an explicit solution to the Ditkin-Prudnikov problem.Constr. Approx.53(2021), no. 1, 1-38. 12.S. Yakubovich, A method of composition orthogonality and new sequences of or- thogonal polynomials and functions for non-classical weights.J. Math. Anal. Appl. 499(2021), 125032. 34...

2021

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Krakowski, On certain functions connected with the Bessel functions,Zastos

3.M. Krakowski, On certain functions connected with the Bessel functions,Zastos. Mat.4(1958), 130-141 (in Polish). 4.N.N. Lebedev,Special Functions and Their Applications, Dover, New York,

1958

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Prudnikov, Yu.A

5.A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev,Integrals and Series. Vol. I: Elementary Functions, Vol. II:Special Functions, Gordon and Breach, New York and London, 1986, Vol. III:More Special Functions, Gordon and Breach, New York and London,

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Yakubovich, A class of integral equations and index transformations related to the modified and incomplete Bessel functions

9.S. Yakubovich, A class of integral equations and index transformations related to the modified and incomplete Bessel functions. J. Integral Equations Appl. 22 (2010), no. 1, 141-164. 10.S. Yakubovich, Certain identities, connection and explicit formulas for the Bernoulli and Euler numbers and the Riemann zeta-values,Analysis35(2015), 1, 59-

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Yakubovich, Orthogonal polynomials with ultra-exponential weight functions: an explicit solution to the Ditkin-Prudnikov problem.Constr

11.S. Yakubovich, Orthogonal polynomials with ultra-exponential weight functions: an explicit solution to the Ditkin-Prudnikov problem.Constr. Approx.53(2021), no. 1, 1-38. 12.S. Yakubovich, A method of composition orthogonality and new sequences of or- thogonal polynomials and functions for non-classical weights.J. Math. Anal. Appl. 499(2021), 125032. 34...

2021