Bessel Generating Functions

• Bessel generating functions are analytic series that encapsulate classical, discrete, and deformed Bessel families, enabling unified expansions and orthogonality relations.
• They employ D-module structures and q-analog techniques to derive transmutation, raising/lowering formulas, and addition theorems with tangible combinatorial insights.
• Applications span random matrix theory, integrable systems, and probability, providing a framework for both continuous and discrete models in harmonic analysis.

Bessel generating functions serve as a unifying analytic and algebraic tool for studying classical, discrete, deformed, and q-analog Bessel families, underpinning expansions, addition formulas, orthogonality, and transforms central to harmonic analysis and special function theory. They encode both structural and combinatorial information, finding applications in random matrix theory, combinatorics, integrable systems, and probability.

1. Classical Bessel Generating Functions

The archetypal generating function is the bilateral expansion

$$\exp\bigg[\frac{x}{2}(t - t^{-1})\bigg] = \sum_{n=-\infty}^{\infty} J_n(x)\,t^n$$

realized as the unique holomorphic solution of a holonomic system arising from a Weyl algebra quotient ("Bessel module") (Chiang et al., 2023). For arbitrary order $\nu$,

$$t^{-\nu}\exp\bigg[\frac{x}{2}(t - t^{-1})\bigg]\sim \sum_{n=-\infty}^{\infty} J_{n+\nu}(x)\,t^n$$

where the equivalence denotes Borel resummation in non-integer order. The generating kernel can be interpreted as the image of a D-linear intertwiner mapping a cyclic vector in the module to elementary functions, with explicit transmutation (raising/lowering) formulas producing contiguous relations. Analogous forms exist for $I_\nu$, $Y_\nu$, and $K_\nu$: $$t^{-\nu}\exp\left[\frac{x}{2}(t+t^{-1})\right] \sim \sum_n I_{n+\nu}(x)t^n, \quad \exp\left[\frac{x}{2}(t - t^{-1})\right] \sim \sum_n Y_{n+\nu}(x)t^n$$ Integral representations—e.g., the Schl\"afli–Sonine formula

$$J_\nu(x) = \frac{1}{2\pi i}\int_{-\infty}^{(0+)} t^{-\nu-1}\exp\left[\frac{x}{2}(t-t^{-1})\right] dt$$

arise naturally from the module structure and underpin inversion and orthogonality (Chiang et al., 2023).

2. Discrete, Difference, and $q$-Analog Generating Functions

Discrete Bessel functions $B_n^{(N)}(m)$ are constructed by discretizing the classical generating function via a sum over $N$ equispaced angles,

$$B_n^{(N)}(m) = \frac{1}{N}\sum_k \exp\bigl(m\sin\varphi_k\bigr)\left[C_n \cos(n\varphi_k) - S_n\sin(n\varphi_k)\right]$$

yielding $N$-point transforms that closely approximate $J_n(m)$ for $n+|m|<N$ and preserve discrete analogues of linear and quadratic identities (e.g., Graf's addition theorem) (Uriostegui et al., 2020).

The $q$-analogues, most notably the Hahn–Exton $q$-Bessel functions,

$$J_\nu(z;q) = \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} \left(\frac{z}{2}\right)^\nu {}_1\phi_1\left(0; q^{\nu+1}; q, -\frac{z^2}{4}\right)$$

yield combinatorial generating functions for skew shapes as ratios $J_{\nu+1}/J_\nu$, which are rational in $q$ with positive coefficients (Kim et al., 2020). The bounded-diagonal case is refined via $q$-Lommel polynomials, and continued fraction expansions are connected to moments of both type $R_I$ and standard orthogonal polynomials, analyzed via the Flajolet–Viennot theory.

Difference Bessel generating functions are obtained from delay-difference realizations of the Bessel module,

$$\Bigl[\frac{1}{2}(t-t^{-1})+1\Bigr]^x = \sum_{n=-\infty}^\infty J_n^\Delta(x)\,t^n$$

with analogous integral and recurrence structure (Chiang et al., 2023).

3. Generalizations: Higher Bessel and Deformed Generating Functions

The higher Bessel ("$N$-Bessel") generating function

$$\Phi^{(N)}(z) = \sum_{n=0}^\infty \frac{z^n}{(n!)^N}$$

arises as the unique solution of $(\theta_x^N - x)\psi(x)=0$ and admits both power-series and multi-contour integral representations. Product formulas introduce a mixing kernel

$$K_N(x,y|z) = \sum_{j,k\geq0}\binom{j+k}{k}^N \frac{x^j y^k}{z^{j+k}}$$

interpreted as the period of a Landau–Ginzburg hypersurface and related to Buchstaber–Rees polynomials, which define $N$-valued group laws (Gaiur et al., 5 May 2024).

Deformed generating functions under Dunkl operator calculus yield, e.g.,

$$G^{(\mu)}(x,t) = \sum_{n=-\infty}^{\infty} J_n^{(\mu)}(x)t^n = E_\mu\left(\frac{x}{2}(t-1/t)\right),$$

where $E_\mu$ is the deformed exponential sum, and Poisson-type kernels lead to further confluent hypergeometric deformations, recovering the classical case as $\mu\to 0$ (Zahaf et al., 2012).

4. Bessel Polynomial and Associated Function Generating Functions

Generating functions for Bessel polynomials $q_n(z)$ and associated Bessel functions $B_{\ell,m}(x)$ permit multiple structural forms:

• The standard Bessel-polynomial generating function:

$$\sum_{n=0}^\infty q_n(z) \frac{t^n}{n!} = \exp\left(\frac{z}{1-t}\right)\frac{1}{1-t}$$

• Multiple-sum (multinomial) identities relating products of Bessel polynomials to sums over their arguments; probabilistic approaches use the generalized inverse Gaussian law to obtain moment representations and collapse multinomial expansions (Lévêque et al., 2012).
• Associated Bessel functions, with three distinct exponential generating functions adapted to the symmetries and reducibilities of their two-parameter indices, can each be given by explicit exponential/radical or contour-integral form—recovering standard polynomial and Laguerre generating functions in special cases (Fakhri et al., 2012).

5. Generating Functions in Probability, Random Matrices, and Integrable Systems

The Bessel point process, a determinantal process with kernel

$$K_\alpha^{\mathrm{Be}}(x,y) = \frac{\sqrt{x} J_{\alpha+1}(\sqrt{x}) J_\alpha(\sqrt{y}) - J_\alpha(\sqrt{x}) J_{\alpha+1}(\sqrt{y})}{2(x-y)}$$

admits a joint probability generating function over $k$ intervals as a Fredholm determinant: $$F(x,s) = \det\left[I - \sum_{j=1}^k(1-s_j)\chi_{(x_{j-1},x_j)} K_\alpha^{\mathrm{Be}} \chi_{(x_{j-1},x_j)}\right]_{L^2(\mathbb{R}_+)}$$ which is equivalently the tau function for a system of $k$ coupled Painlevé V equations, derived via a Riemann–Hilbert Lax pair construction. This formalism underpins probability generating functions for gap probabilities, particle ratios, and large-$n$ Hankel determinant asymptotics near the hard edge of random matrices (Charlier et al., 2017).

6. Algebraic, Combinatorial, and D-module Foundations

The algebraic structure of holonomic D-modules (quotients of the Weyl algebra by Bessel operators) determines the existence, form, and uniqueness of Bessel generating functions. Transmutation and intertwining operators encode contiguous and ladder relations, unifying a broad swath of generating functions from classical to difference and deformed types (Chiang et al., 2023). Product formulas for higher Bessel generating functions are intimately related to periods of algebraic hypersurfaces and generalized Frobenius expansions, with conjectured unimodality and real-rootedness of associated palindromic polynomials (Gaiur et al., 5 May 2024).

In combinatorics, q-Bessel and $q$-Lommel generating functions enumerate connected skew shapes and are central in continued fraction expansions interpreted via weighted Motzkin paths, using orthogonal polynomial and lattice path methods (Kim et al., 2020).

7. Recurrence Relations, Bell Polynomials, and Umbral Formalism

Powers of Bessel and modified Bessel functions generate families of polynomials $P_n^{(m)}(v)$, whose exponential generating functions are $(I_v(z))^m$. Explicit expressions via Bell polynomials and the Bessel zeta function

$$P_n^{(m)}(v) = B_n(a_1,\dots,a_n), \quad a_r = (-1)^{r-1}(r-1)!\zeta_v(2r)m$$

enable combinatorial and asymptotic enumeration, while recurrences are interpreted probabilistically via moment expansions of the symmetric beta distribution. The Cholewinski umbral formalism reveals binomial-type convolution properties for these polynomial families (Moll et al., 2013).

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Key references:

• Discrete and transform-based structure (Uriostegui et al., 2020)
• D-module unification (Chiang et al., 2023)
• Higher order/group law connections (Gaiur et al., 5 May 2024)
• $q$-analogs and combinatorics (Kim et al., 2020)
• Probabilistic and polynomial expansions (Moll et al., 2013, Lévêque et al., 2012)
• Associated functions (Fakhri et al., 2012)
• Deformed/difference families (Zahaf et al., 2012)
• Applications in random matrix theory/Painlevé integrability (Charlier et al., 2017)