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Approximating the coefficients of the Bessel functions

Andrew Yao

TL;DR

The paper analyzes the asymptotics of Bessel functions $J_a^{\mathcal{R}(\theta)}(x)$ tied to root systems $A^{N-1}$, $B^N$, $C^N$, and $D^N$ by studying Dunkl operators and their bilinear form. It establishes equivalence between the growth of coefficients in logarithmic inputs and the asymptotics of the Dunkl bilinear form under large-$N$ scaling in several regimes, and derives leading-order terms for the corresponding bilinear forms across types A, BC, and D. A graded-ring framework of operators is developed to connect these asymptotics with eigenfunctions, enabling detailed expansions for Bessel coefficients and enabling weak-convergence results to free or rectangular free convolutions. The results generalize prior limits-probability frameworks and provide combinatorial and gamma-function expressions that describe the dominant contributions, with implications for universality and representation-theoretic structure of Dunkl-type harmonic analysis on root systems.

Abstract

For the type A, BC, and D root systems, we determine equivalent conditions between the coefficients of an exponential holomorphic function and the asymptotic values taken by the Dunkl bilinear form when one of its entries is the function. We establish these conditions over the $|θN| \rightarrow\infty$ regime for the type A and D root systems and over the $|θ_0 N|\rightarrow \infty, \frac{θ_1}{θ_0 N}\rightarrow c\in\mathbb{C}$ regime for the type BC root system. We also generalize existing equivalent conditions over the $θN \rightarrow c\in\mathbb{C}$ regime for the type A root system and over the $θ_0 N\rightarrow c_0\in\mathbb{C}, \frac{θ_1}{θ_0 N}\rightarrow c_1\in\mathbb{C}$ regime for the type BC root system and prove new equivalent conditions over the $θN \rightarrow c\in\mathbb{C}$ regime for the type D root system. Furthermore, we determine the asymptotics of the coefficients of the Bessel functions over the regimes that we have mentioned.

Approximating the coefficients of the Bessel functions

TL;DR

The paper analyzes the asymptotics of Bessel functions tied to root systems , , , and by studying Dunkl operators and their bilinear form. It establishes equivalence between the growth of coefficients in logarithmic inputs and the asymptotics of the Dunkl bilinear form under large- scaling in several regimes, and derives leading-order terms for the corresponding bilinear forms across types A, BC, and D. A graded-ring framework of operators is developed to connect these asymptotics with eigenfunctions, enabling detailed expansions for Bessel coefficients and enabling weak-convergence results to free or rectangular free convolutions. The results generalize prior limits-probability frameworks and provide combinatorial and gamma-function expressions that describe the dominant contributions, with implications for universality and representation-theoretic structure of Dunkl-type harmonic analysis on root systems.

Abstract

For the type A, BC, and D root systems, we determine equivalent conditions between the coefficients of an exponential holomorphic function and the asymptotic values taken by the Dunkl bilinear form when one of its entries is the function. We establish these conditions over the regime for the type A and D root systems and over the regime for the type BC root system. We also generalize existing equivalent conditions over the regime for the type A root system and over the regime for the type BC root system and prove new equivalent conditions over the regime for the type D root system. Furthermore, we determine the asymptotics of the coefficients of the Bessel functions over the regimes that we have mentioned.

Paper Structure

This paper contains 48 sections, 143 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. The regime
  4. Related works
  5. Weak convergence to the free convolution
  6. Uniform convergence of the Bessel functions
  7. Coefficients of terms with all odd degrees in the type D Bessel function
  8. Applying a graded ring of operators to a graded vector field
  9. Equivalence conditions for non-exponential holomorphic functions
  10. Paper organization
  11. Acknowledgments
  12. Basic definitions and notation
  13. Dunkl operators
  14. Partitions
  15. Non-crossing partitions
  16. ...and 33 more sections

Key Result

Theorem 1.1

Assume that $\lim_{N\rightarrow\infty} |\theta N|=\infty$, $\lim_{N\rightarrow\infty} |\theta_0 N|=\infty$, and $\lim_{N\rightarrow\infty} \frac{\theta_1}{\theta_0 N} = c\in\mathbb{C}$. (A): Suppose $F^A_N(x_1,\ldots,x_N) = \exp\left(\sum_{\lambda\in\Gamma_N} c_\lambda(N) p_\lambda\right)\in\mathbb{ (B): Suppose $F^{BC}_N(x_1,\ldots,x_N) = \exp\left(\sum_{\lambda\in\Gamma_{\text{even};\,N}} c_{\la

Theorems & Definitions (291)

  • Theorem 1.1
  • proof
  • Lemma 1.2
  • Remark 1.3
  • proof : Proof of \ref{['lemma:mainrestate']}
  • Theorem 1.4: rosler_positivity
  • Remark 1.5
  • Conjecture 1.6
  • Corollary 1.7
  • Corollary 1.8
  • ...and 281 more