[Paper Review] On fractional powers of Bessel operators

Elina Shishkina, С. М. Ситник|arXiv (Cornell University)|Mar 7, 2017

Differential Equations and Boundary ProblemsMathematics19 citations

TL;DR

This paper introduces explicit integral representations for fractional powers of the Bessel differential operator $ B_{ u} = D^2 + \frac{\nu}{x}D $, avoiding reliance on integral transforms like the Hankel or Mellin transforms. It establishes key properties including group structure, generalized Taylor formulas using hypergeometric functions, and connections to Wright and Mittag-Leffler functions, offering a foundation for further generalizations in fractional calculus and PDEs.

ABSTRACT

This paper was published in the special issue of the Journal of Inequalities and Special Functions dedicated to Professor Ivan Dimovski's contributions to different fields of mathematics: transmutation theory, special functions, integral transforms, function theory etc. In this paper we study fractional powers of the Bessel differential operator. The fractional powers are defined explicitly in the integral form without use of integral transforms in its definitions. Some general properties of the fractional powers of the Bessel differential operator are proved and some are listed. Among them are different variations of definitions, relations with the Mellin and Hankel transforms, group property, generalized Taylor formula with Bessel operators, evaluation of resolvent integral operator in terms of the Wright or generalized Mittag--Leffler functions. At the end, some topics are indicated for further study and possible generalizations. Also the aim of the paper is to attract attention and give references to not widely known results on fractional powers of the Bessel differential operator.

Motivation & Objective

• To provide explicit, transform-free integral definitions for fractional powers of the Bessel differential operator $ B_{\nu} $, overcoming limitations of implicit definitions via Hankel transforms.
• To unify and extend existing fractional calculus frameworks by incorporating Bessel operators with standard notation and explicit integral forms.
• To establish foundational properties such as group structure, resolvent evaluation via Wright and Mittag-Leffler functions, and generalized Taylor expansions.
• To highlight underappreciated results and provide comprehensive references to promote further research in fractional Bessel operators and their applications.
• To enable new applications in PDEs, transmutation theory, and commuting differential operators through explicit kernel representations.

Proposed method

• Define right- and left-sided fractional Bessel integrals $ B_{\nu,b-}^{-\alpha} $ and $ B_{\nu,a+}^{-\alpha} $ using Gauss hypergeometric functions $ {}_2F_1 $ as kernels.
• Construct explicit integral operators for fractional powers of $ B_{\nu} $ using the form $ \frac{1}{\Gamma(2\alpha)} \int_{x}^{b} \left( \frac{y^2 - x^2}{2y} \right)^{2\alpha-1} {}_2F_1(\cdots) f(y) \, dy $, valid for $ \alpha > 0 $.
• Utilize known integral representations from [7] and [13]–[14] to unify definitions with classical fractional calculus notation.
• Establish generalized Taylor formulas by expressing function expansions in terms of $ B^{i-1}f $ and $ DB^{i-1}f $ evaluated at boundary points, with remainder terms as fractional integrals.
• Relate the resolvent operator to Wright and generalized Mittag-Leffler functions through integral kernel analysis.
• Extend results to related operators such as the Clifford-type Bessel operator $ C_\nu $, and discuss connections to transmutation theory and commuting differential operators.

Experimental results

Research questions

• RQ1 How can fractional powers of the Bessel operator $ B_{\nu} $ be defined explicitly without relying on integral transforms?
• RQ2 What are the group properties and structural identities satisfied by the fractional powers of $ B_{\nu} $?
• RQ3 How can generalized Taylor expansions be constructed using Bessel operators in place of derivatives, with fractional integral remainders?
• RQ4 What is the connection between the resolvent of $ B_{\nu} $ and special functions such as the Wright or generalized Mittag-Leffler functions?
• RQ5 In what ways can the explicit fractional Bessel operators be generalized to higher-order or mixed-type differential operators, and how do they relate to existing fractional PDEs?

Key findings

• Fractional powers of the Bessel operator $ B_{\nu} $ are explicitly defined via integral kernels involving Gauss hypergeometric functions, avoiding dependence on Hankel or Mellin transforms.
• The generalized Taylor formula (Theorem 8.1) expresses a function as a finite sum of Bessel operator actions at a boundary point, plus a remainder term involving $ B_{\nu,b-}^{-k}(B^k f) $, with coefficients in terms of hypergeometric functions.
• For the Clifford-type Bessel operator $ C_\nu $, a similar generalized Taylor formula (Theorem 8.2) is derived, incorporating additional terms involving $ x^{-\nu} $ and derivatives.
• The resolvent of $ B_{\nu} $ is evaluated in terms of Wright or generalized Mittag-Leffler functions, establishing a direct link to special functions in fractional calculus.
• When $ \nu = 1 $, the kernels simplify to classical Legendre functions, and for $ \alpha = -\frac{1}{2} $, the fractional powers yield generalized elliptic functions, enabling explicit definitions of Bessel Riesz transforms.
• The framework allows for generalizations to sums of fractional Bessel operators, including B–Riesz potentials in elliptic, hyperbolic, and ultrahyperbolic settings, and supports new classes of transmutation operators and commuting differential operators in quantum mechanics.