Multivariate Bessel Functions

• Multivariate Bessel functions are generalizations of classical Bessel functions, constructed via variable separation and multidimensional operators to solve complex partial differential equations.
• They admit elegant integral representations and series expansions that simplify multidimensional analysis in random matrix theory, signal processing, and various physical models.
• Applications range from spectral theory and stochastic dynamics to digital filter design and quantum mechanics, illustrating their broad utility in modern research.

Multivariate Bessel functions are generalizations of classical Bessel functions to the context of multiple variables and indices, naturally arising in the paper of integrable systems, random matrix theory, multivariate statistics, mathematical physics, and special functions. Their analytic structure encompasses products of Bessel functions, generalized integral representations, orthogonal polynomial connections, and a deep linkage to multidimensional Hankel transforms and partial differential equations. Several rigorous formulations, identities, and applications characterize their use for multidimensional analysis, stochastic dynamics, and spectral theory.

1. Fundamental Definitions and Constructions

Multivariate Bessel functions are not obtained by naive extension of the one-dimensional theory; their construction relies on variable separation in adapted coordinate systems (cylindrical, spherical, parabolic) and products of one-dimensional Bessel functions as bases for multivariate differential operators (Zakharov, 18 Oct 2024). For instance, in $n$ dimensions: $$J_{(a_1, ..., a_n)}(x_1, ..., x_n) = \prod_{j=1}^{n} J_{a_j}(x_j),$$ where each $J_{a_j}(x_j)$ solves the standard Bessel equation. These multivariate functions solve multidimensional analogues of Bessel-type partial differential equations, such as: $$\sum_{j=1}^n D_{a_j;x_j} J_{(a_1,...,a_n)} = -n J_{(a_1,...,a_n)},$$ with $D_{a;x}[f(x)] = \frac{d}{dx}f(x) + \frac{a}{x}f(x)$, encoding the multivariate generalization of the Bessel operator.

Unified generalizations extend the family to products, Bessel-Clifford, and spherical Bessel functions via a generalized Pochhammer series: $$\mathcal{G}(b,c)(\mathbf{z};p) = \sum_{\mathbf{k}\in\mathbb{N}_0^n} \frac{(-b)^{|\mathbf{k}|} (c;p)_{2|\mathbf{k}|+v}}{\mathbf{k}! \Gamma(v+2|\mathbf{k}|+1)} \prod_{j=1}^n \left(\frac{z_j}{2}\right)^{2k_j+v_j},$$ yielding integral representations and generating functions that facilitate analytic manipulation in higher-dimensional settings (Yaşar et al., 2016).

In matrix and representation-theoretic contexts, multivariate Bessel functions are constructed as integrals over Gelfand–Tsetlin polytopes, encoding branching processes, eigenvalue distributions, and relations to symmetric polynomials (e.g., Macdonald, Heckman–Opdam functions) (Sun, 2016).

2. Integral Representations, Series, and Transform Methods

Multivariate Bessel functions frequently admit elegant integral representations that generalize classical results. For modified Bessel functions of order zero, the representation

$$I_0(t) = \frac{1}{\pi}\int_0^\pi e^{t \cos\theta} d\theta$$

extends to products over $n$ variables, providing the factorization $I_0(x)^n$ fundamental in reducing multidimensional integrals, such as Mahler measures, to single integrals involving powers of $I_0$ (Glasser, 2012). This is exploited to obtain expressions for the Mahler measure of the $n$-variate hypercubic polynomial: $$J_n(z) = \int_0^\infty \frac{dx}{x}\left[e^{-x} - e^{-z x} I_0(x)^n\right].$$

In the theory of Hankel transforms, the multivariate analogue employs kernel functions as products: $$S_{(\nu_1, ..., \nu_n)}[f; (k_1, ..., k_n)] = \int_{[0,\infty)^n} f(\mathbf{r}) J_{(\nu_1, ..., \nu_n)}(k_1 r_1, ..., k_n r_n) \prod_{j=1}^n r_j dr_j,$$ where differentiation in $r_j$ translates to multiplication by $-k_j^2$ in transform space, generalizing the Fourier property (Zakharov, 18 Oct 2024). Differential operators and transforms interact via: $$P(-k_1^2, ..., -k_n^2) J_{(\nu_1,...,\nu_n)} = P(D_{a_1;r_1},...,D_{a_n;r_n}) J_{(\nu_1,...,\nu_n)},$$ for any polynomial $P$.

Generalized Bessel functions and multivariate Anger functions further enrich the family, accommodating arbitrary complex orders through integral representations: $$A_\alpha(\mathbf{x},\mathbf{y}) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \exp\left(i[\alpha\theta - \sum_{k=1}^m (x_k \sin k\theta + y_k \cos k\theta)]\right) d\theta,$$ allowing summation identities such as the generalized Lerche–Newberger formula for harmonic sums (Kuklinski et al., 2021).

3. Differential Equations and Analytical Properties

Multivariate Bessel functions solve multidimensional systems of PDEs generalizing the classical Bessel equation. For $m$-dimensional generalized Bessel functions (GBFs) with indices $p=(p_1,...,p_m)$, the canonical form

$$J_n^{(p)}(x) = \frac{1}{2\pi}\int_{-\pi}^\pi e^{i[n\theta - \sum_{k=1}^m x_k \sin(p_k \theta)]} d\theta,$$

enables the derivation of coupled PDEs: $$(n + 2y) f_{xx} - y f_{yy} - \frac{x}{2} f_{xy} - f_y = 0,$$ and Schrödinger-type first-order equations for mixed-type GBFs: $$n J_n = i \sum_{k=1}^m k[x_k J_{n,y_k} - y_k J_{n,x_k}],$$ highlighting the rich interplay between Fourier analysis, group theory, and orthogonal polynomial theory (1908.11683).

Unified approaches extend to operator monomiality principles, generating recurrence relations and explicit formulas in the multivariate setting (Yaşar et al., 2016).

4. Connections to Random Matrix Theory, Statistical Physics, and Probability

Multivariate Bessel functions are central to the description of eigenvalue correlations, spacing distributions, and transition densities for matrix ensembles in probability and statistical mechanics.

For $\beta$-Wishart and $\beta$-Jacobi ensembles, joint eigenvalue densities are expressed as multivariate Bessel ensembles, facilitating explicit characterization of multilevel processes and providing exact connections with Heckman–Opdam measures and symmetric function theory (Sun, 2016). In stochastic dynamics, multivariate Bessel processes defined on Weyl chambers model Calogero–Moser–Sutherland systems, with collision times, fluctuation behavior, and freezing limits precisely described via Bessel function representations (Andraus et al., 2018, Voit, 2018, Hufnagel et al., 2023). For instance, the transition density kernel

$$K_t(x,E) = c_{k,t} \exp\left(-\frac{|x|^2+|y|^2}{2t}\right) J_k\left(-\frac{x}{\sqrt{t}},\frac{y}{\sqrt{t}}\right) w_A(y) dy$$

encodes both the combinatorics of the underlying reflection group and the interaction strength (multiplicity) (Kornyik et al., 2019).

The characteristic functions for multivariate elliptical distributions are written in terms of Bessel function integrals: $$\varphi(u^2) = C_n 2^{n-2} u^{2-n} \int_0^\infty r^{n-1} J_{n-2}(ru) g(r^2) dr,$$ providing unified closed forms for distributions such as multivariate Student-$t$, Cauchy, Laplace, and stable laws, underpinning statistical models with rotational symmetry (Yin et al., 2021).

5. Special Cases, Orthogonality, and Representation Theory

Special instances—such as three-particle integrals in quantum mechanics, lattice sums, and spanning tree generating functions—reduce to single integrals or rapidly convergent sums involving multivariate products or powers of $I_0$ or $J_0$ (Glasser, 2012, Frolov et al., 2012). Orthogonality results persist in the multivariate $q$-Bessel setting, where functions such as

$$J_{\mathbf{v}}(x;A;q) = \prod_{j=1}^d J_{v_j-v_{j+1}-1}(q^{v_{j+1}+1}x_j;q)$$

satisfy discrete orthogonality and self-duality identities, connecting directly to $3nj$-symbols and the representation theory of $q$-deformed quantum groups (Groenevelt, 2016). Multivariate $q$-Bessel functions appear as limit cases of multivariate Askey–Wilson polynomials, providing a bridge between hypergeometric orthogonal functions and coupling coefficients in representation theory.

Explicit closed-form evaluations, moment formulas, and bifurcation surfaces in the asymptotics of GBFs yield analytic insight into the oscillatory regimes and critical behavior in multidimensional parameter spaces (1908.11683).

6. Applications, Implications, and Universality

Multivariate Bessel functions unify multidimensional signal processing, mathematical physics, harmonic analysis, and high-dimensional probability. In digital signal processing, multivariate extensions of the Kaiser window (based on $I_0$) promise new multidimensional filter designs (Yaşar et al., 2016). Their appearance in collision phenomena (e.g., Dyson models) is quantified: the Hausdorff dimension of boundary hitting times is exactly

$$\max\bigg(0, \frac{1}{2} - \min_{\alpha \in R_+} k(\alpha)\bigg),$$

universal across space dimension and particle number (Hufnagel et al., 2023).

Physical applications abound in quantum systems, plasma physics, laser optics, and electromagnetics, where generalized summation formulas (such as the multivariate Lerche–Newberger identity) provide closed-form solutions for sums over energy or transition rates in frequency-modulated environments (Kuklinski et al., 2021).

7. Summary Table of Key Multivariate Bessel Function Types

Type	Core Definition	Primary Domains of Application
Product Type	$$J_{(a_1,..., a_n)}(x_1,..., x_n) = \prod_j J_{a_j}(x_j)$$	PDEs, transforms, signal processing
Unified (Pochhammer)	Series over multi-indices (see above)	Orthogonal polynomials, transforms
Generalized/Anger Functions	Integral with sinusoidal phase (see above)	Multi-tone modulation, quantum systems
Matrix Bessel	Integral on symmetric cones, Gelfand–Tsetlin polytopes	Random matrices, branching processes
$q$-Bessel	Product of Hahn–Exton $q$-Bessel functions (see above)	Quantum group representations
Multivariate GBF	Integral representation with $m$ indices (see above)	Laser physics, PDEs, asymptotics

The multivariate Bessel function encompasses an entire class of special functions, integral transforms, and analytic tools that enable unified treatment of multidimensional analysis, with exact identities, transform properties, and deep connections to physical and probabilistic models. Its universality and structural richness make it indispensable in contemporary research spanning mathematics, physics, and engineering.