[Paper Review] Unified Bessel, Modified Bessel, Spherical Bessel and Bessel-Clifford Functions

Banu Yılmaz Yaşar, Mehmet Ali Özarslan|arXiv (Cornell University)|Apr 2, 2016

Mathematical functions and polynomials39 references3 citations

TL;DR

This paper introduces a unified family of Bessel-type functions—encompassing Bessel, modified Bessel, spherical Bessel, and Bessel-Clifford functions—using the generalized Pochhammer symbol. By leveraging this generalization, the authors derive integral representations, transforms (Laplace, Mellin), recurrence relations, and differential equations, providing a comprehensive framework that subsumes known special functions and yields new results in special function theory.

ABSTRACT

In the present paper, unification of Bessel, modified Bessel, spherical Bessel and Bessel-Clifford functions via the generalized Pochhammer symbol [ Srivastava HM, Cetinkaya A, Kıymaz O. A certain generalized Pochhammer symbol and its applications to hypergeometric functions. Applied Mathematics and Computation, 2014, 226 : 484-491] is defined. Several potentially useful properties of the unified family such as generating function, integral representation, Laplace transform and Mellin transform are obtained. Besides, the unified Bessel, modified Bessel, spherical Bessel and Bessel-Clifford functions are given as a series of Bessel functions. Furthermore, the derivatives, recurrence relations and partial differential equation of the so-called unified family are found. Moreover, the Mellin transform of the products of the unified Bessel functions are obtained. Besides, a three-fold integral representation is given for unified Bessel function. Some of the results which are obtained in this paper are new and some of them coincide with the known results in special cases.

Motivation & Objective

• To unify Bessel, modified Bessel, spherical Bessel, and Bessel-Clifford functions into a single generalized family using the generalized Pochhammer symbol.
• To derive fundamental analytical properties such as generating functions, integral representations, and Laplace and Mellin transforms for the unified function family.
• To establish recurrence relations, derivatives, and partial differential equations for the unified family.
• To demonstrate that known special functions emerge as special cases of the unified family.
• To provide a three-fold integral representation and evaluate Mellin transforms of products of the unified functions.

Proposed method

• The unified function is defined via a generalized hypergeometric series using the generalized Pochhammer symbol, extending classical Bessel-type functions.
• The generalized Pochhammer symbol is constructed from the extended Beta function, allowing for a broader parameter space.
• Integral representations, including a three-fold integral, are derived using integral transforms and generalized hypergeometric identities.
• Laplace and Mellin transforms of the unified function are computed using integral definitions and properties of the generalized Pochhammer symbol.
• Recurrence relations and differential equations are derived by manipulating series representations and applying operator identities.
• The unified function is expressed as a series of classical Bessel functions, enabling connections to known function classes.

Experimental results

Research questions

• RQ1 Can Bessel, modified Bessel, spherical Bessel, and Bessel-Clifford functions be embedded within a single generalized function family using the generalized Pochhammer symbol?
• RQ2 What are the integral, transform, and differential properties of this unified function family?
• RQ3 How do known special functions arise as limiting cases of the unified family?
• RQ4 What is the structure of the Mellin transform of products of the unified Bessel functions?
• RQ5 Can the unified function be represented as a series of classical Bessel functions?

Key findings

• The unified Bessel function is expressed as a series of classical Bessel functions, establishing a direct link between the generalized and classical forms.
• A three-fold integral representation is derived for the unified Bessel function, generalizing known integral forms.
• The Mellin transform of the product of two unified Bessel functions is computed, yielding a generalized hypergeometric function.
• The unified function satisfies a second-order partial differential equation, generalizing the classical Bessel differential equation.
• The Laplace and Mellin transforms of the unified function are explicitly evaluated using the generalized Pochhammer symbol.
• Several known results for classical Bessel functions are recovered as special cases when parameters reduce to standard values.