[Paper Review] The integral representations of the q-Bessel-Macdonald functions

V. -B. K. Rogov|ArXiv.org|Jan 31, 2001

Advanced Mathematical Identities4 citations

TL;DR

This paper establishes new integral representations for $q$-Bessel-Macdonald functions ($q$-BMF) of types 1, 2, and 3 using classical double integrals, overcoming limitations of $q$-lattice-based Jackson integrals. By constructing explicit integral formulas involving $q$-exponential and $q$-hypergeometric functions, the authors derive representations that generalize the classical Macdonald function integral in the $q \to 1$ limit, enabling analytic continuation and harmonic analysis on quantum Lobachevsky space.

ABSTRACT

The q-Bessel-Macdonald functions of kinds 1, 2 and 3 are considered. Their representations by classical integral are constructed.

Motivation & Objective

• To overcome the limitations of $q$-lattice-based Jackson integrals in harmonic analysis on quantum Lobachevsky space.
• To construct classical double integral representations for $q$-Bessel-Macdonald functions of all three types (1, 2, 3).
• To generalize the classical integral representation of the Macdonald function to the $q$-deformed case.
• To ensure analytic continuation of $q$-BMF beyond the convergence radius of their series expansions.

Proposed method

• Derives $q$-modified Bessel functions $I_{\nu}^{(j)}((1-q^2)z; q^2)$ from basic hypergeometric series and $q$-gamma functions.
• Introduces $q$-Bessel-Macdonald functions $K_{\nu}^{(j)}$ via a $q$-Wronskian construction using $I_{\nu}^{(j)}$ and $I_{-\nu}^{(j)}$.
• Constructs auxiliary functions $\xi_{\eta}^{(\delta)}(s)$ as $q$-exponential or $q$-hypergeometric series to model kernel components.
• Derives double integral representations by integrating products of $\xi_{\eta}^{(\delta)}$ functions over the complex plane with $q$-exponential weights.
• Establishes convergence and meromorphic continuation by analyzing $q$-series and $q$-binomial identities.
• Verifies the $q \to 1$ limit to recover the classical Macdonald function integral representation.

Experimental results

Research questions

• RQ1 Can $q$-Bessel-Macdonald functions of types 1, 2, and 3 be represented by classical double integrals, independent of $q$-lattice structures?
• RQ2 How do the $q$-modified Bessel functions $I_{\nu}^{(j)}$ and $K_{\nu}^{(j)}$ relate to classical Bessel functions in the $q \to 1$ limit?
• RQ3 What is the role of the $q$-Wronskian in constructing a fundamental solution system for the $q$-difference equation?
• RQ4 How can the double integral representation be analytically continued beyond the radius of convergence of the $q$-series?

Key findings

• The $q$-Bessel-Macdonald function $K_{\nu}^{(j)}(2(1-q^2)|z|, q^2)$ admits a double integral representation via $\xi_{\eta}^{(\delta)}$ functions and $q$-exponentials.
• The integral formula (5.9) and (5.10) are valid for $\delta=2,0,1$, corresponding to $j=1,2,3$, respectively, with explicit dependence on $A_\nu$ and $\Gamma_{q^2}(\nu+1)$.
• For $\delta < 2$, the integral is computed term-by-term using orthogonality of $e^{in\phi}$, yielding a $q$-Bessel function as the result.
• In the limit $q \to 1^-$, the $q$-integral representation reduces to the classical integral form of the Macdonald function.
• The representation remains valid for $r\rho < 1/(1-q^2)$ and extends meromorphically to the full complex plane via analytic continuation.
• The $q$-Wronskian of $I_{\nu}^{(j)}$ and $K_{\nu}^{(j)}$ is non-vanishing, confirming they form a fundamental solution system for the $q$-difference equation.