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

Abstract: We continue to investigate properties of the function $M_ν(z)$ which is associated with the Macdonald function $K_ν(z)$ 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 $ \hatρν(x)= 2 x^{ν/2} Mν\left(2\sqrt x\right), x >0$ are investigated.

• The paper presents the derivation of differential recurrences and integral representations that extend classical Bessel and Macdonald functions.
• It employs spectral theory and transform techniques to link the Macdonald-type function with Laguerre and multiple orthogonal polynomials.
• The work establishes practical frameworks for numerical evaluation in spectral problems and lays foundations for future asymptotic studies.

Macdonald-type Functions: Index Transforms, Orthogonal Polynomials, and Analytical Structure

Introduction

This paper presents a comprehensive investigation of the Macdonald-type function $M_\nu(z)$, establishing its foundational analytical properties, differential recurrence relations, and, crucially, its connections with index transforms and the theory of multiple orthogonal polynomials. The analysis leverages integral representations, spectral theory of differential operators, explicit Mellin-Barnes and Laplace transforms, and connections with classical special functions, extending the landscape of Bessel- and Macdonald-related objects in both function and polynomial spaces.

Properties and Differential Structure of $M_\nu(z)$

The function $M_\nu(x)$ is defined through the integral

$$M_{\nu}(x)= \int_{0}^\infty e^{-x\cosh u} \sinh(\nu u) du,$$

which can be viewed as a natural companion to the Macdonald (modified Bessel) function $K_\nu(x)$. Unlike $K_\nu(x)$, which solves a homogeneous Bessel differential equation, $M_\nu(x)$ satisfies the inhomogeneous equation

$$x^2 y^{\prime\prime}(x) + x y^{\prime}(x) - (x^2+\nu^2) y(x)= \nu e^{-x}.$$

This singles out $M_\nu$ as an analytically richer object, blending classical Bessel theory with forced oscillatory behavior via the nonhomogeneous term. The paper derives a system of differential and recurrence relations for $M_\nu$ and its derivatives, including mixed index and argument recursions. For instance,

$M_\nu(z)$0

$M_\nu(z)$1

These relations parallel, yet significantly extend, the classic recurrences of Bessel and Macdonald functions.

Spectral and Integral Representations

Extensive use is made of spectral theory, notably through the operator

$M_\nu(z)$2

with $M_\nu(z)$3 and $M_\nu(z)$4 as (generalized) eigenfunctions with different spectral properties due to the inhomogeneous nature of the equation for $M_\nu(z)$5. This operator framework facilitates the derivation of higher-order recurrences and the identification of associated polynomials through manipulation of the operator's iterates.

Integral representations are central. Not only is a Mellin-Barnes formula developed for $M_\nu(z)$6, but the paper systematically relates $M_\nu(z)$7 to $M_\nu(z)$8 and to squares and products of "scaled" Macdonald functions $M_\nu(z)$9 through Laplace and Mellin transforms. For example,

$M_\nu(x)$0

Integral manipulation (including differentiation under the integral and application of Fubini's theorem) provides various representations—integral, series, and convolutional—for the derivatives and mixed-index components.

Relations with Laguerre and Multiple Orthogonal Polynomials

A core achievement is connecting $M_\nu(x)$1 to associated Laguerre polynomials. Utilizing the Rodrigues formula for $M_\nu(x)$2 and differential relations, the work produces hybrid recurrences linking $M_\nu(x)$3 and its derivatives with weighted Laguerre polynomials and generalized moments. These yield convolution-type integral formulas and multi-term recurrences for families of functions $M_\nu(x)$4 of the form

$M_\nu(x)$5

which bridge $M_\nu(x)$6 and orthogonal polynomial theory.

The Macdonald-type weight family $M_\nu(x)$7 emerges as a central object for constructing systems of multiple orthogonal polynomials. Explicit and recursive constructions of these polynomials—Type I and Type II—are explored, with in-depth analysis of their moments, Rodrigues formulas, and recurrence structures, including the appearance of third-order recursions indicative of multiple orthogonality.

Analytical and Transform Properties

The treatise provides comprehensive transform-theoretic analysis, including:

• Connections of $M_\nu(x)$8 to Kontorovich-Lebedev-type index transforms, both through their kernel properties and associated spectral theory.
• Reciprocal pairs of transforms involving $M_\nu(x)$9 and modified Bessel functions of the first kind, with inversion formulas derived and conditions for their validity in relevant function spaces.
• Mellin-Barnes and Laplace representations, intricately connected to parabolic cylinder functions, Whittaker functions, and generalized hypergeometric functions $$M_{\nu}(x)= \int_{0}^\infty e^{-x\cosh u} \sinh(\nu u) du,$$0.

Integral and series expansions for specific auxiliary functions $$M_{\nu}(x)= \int_{0}^\infty e^{-x\cosh u} \sinh(\nu u) du,$$1 are developed, including convergent series in terms of $$M_{\nu}(x)= \int_{0}^\infty e^{-x\cosh u} \sinh(\nu u) du,$$2 and explicit upper bounds for analytic continuation and norm estimates.

Structural Results and Differential Recurrences for Multiple Orthogonal Polynomials

The paper rigorously formulates the structure of multiple orthogonal polynomials for the Macdonald-type weights, with a dual system corresponding to indices $$M_{\nu}(x)= \int_{0}^\infty e^{-x\cosh u} \sinh(\nu u) du,$$3 and $$M_{\nu}(x)= \int_{0}^\infty e^{-x\cosh u} \sinh(\nu u) du,$$4. Both Type I (polynomial pairs $$M_{\nu}(x)= \int_{0}^\infty e^{-x\cosh u} \sinh(\nu u) du,$$5) and Type II (monic polynomials $$M_{\nu}(x)= \int_{0}^\infty e^{-x\cosh u} \sinh(\nu u) du,$$6) are studied through their orthogonality conditions with respect to the weight vector $$M_{\nu}(x)= \int_{0}^\infty e^{-x\cosh u} \sinh(\nu u) du,$$7.

Main analytical results include:

• Differential formulas generalizing the classical Rodrigues formula to the multiple orthogonal setting:

$$M_{\nu}(x)= \int_{0}^\infty e^{-x\cosh u} \sinh(\nu u) du,$$8

up to normalizing factors.

• Explicit expressions and generating functions for the multiple orthogonal polynomial sequences.
• Recurrence formulas interlacing index and order, and an explicit combinatorial representation for the polynomials in terms of moments and weights.

The moments of the weight measures are expressed in terms of Whittaker and parabolic cylinder functions and, for the modified weight, in terms of infinite series involving generalized hypergeometric functions.

Implications and Outlook

This work materially extends the analysis and analytical apparatus surrounding Macdonald-type functions, placing $$M_{\nu}(x)= \int_{0}^\infty e^{-x\cosh u} \sinh(\nu u) du,$$9 as a rich generalization in the hierarchy of special functions, with direct consequences for spectral theory, index transforms, and the theory of multiple orthogonal polynomials.

Potential theoretical implications include:

• Expansion of the toolkit for index and Laplace transform inversion in applied analysis, particularly integral equations with kernels of Macdonald type.
• New explicit families of multiple orthogonal polynomials amenable to asymptotic and zero-distribution analysis, enabling applications in random matrix theory, moment problems, and approximation theory.
• Development of explicit series, integral, and recursion formulas suited for numerical computation and further study in analytic number theory and special function theory.

On the practical side, the explicit integral representations and bounds allow for rigorous numerical evaluation and estimation in applied spectral problems where Macdonald-type kernels arise, e.g., in mathematical physics and engineering models involving damping or attenuated wave propagation.

Future directions include a deeper study of the zero distribution and asymptotic properties of the associated multiple orthogonal polynomials, the structure of the analogues of the Christoffel-Darboux kernel, and further connections to integrable systems and Painlevé equations.

Conclusion

The paper delivers a thorough analytical framework for the study of the Macdonald-type function $K_\nu(x)$0, elucidating its relationship with classical and generalized Bessel functions, constructing a hierarchy of differential and integral relations, and embedding it within the rich theory of multiple orthogonal polynomials. The results achieved open new avenues in both function theory and the structure of polynomial systems affiliated with non-classical weights, establishing not only foundational properties but also paths for future theoretical and applied research.