Product formulas for the Higher Bessel functions

TL;DR

This paper explores the mathematical structure of higher Bessel functions through product formulas, algebraic hypersurfaces, and special polynomial properties, revealing deep connections with algebraic geometry, number theory, and mirror symmetry.

Contribution

It introduces new product formulas for higher Bessel functions, links them to algebraic hypersurfaces and group laws, and investigates their polynomial roots and motives.

Findings

Zeros of Buchstaber-Rees polynomials define singular loci.

Experimental connections between N-Bessel kernels and palindromic polynomials.

Conjecture on positivity of roots of certain polynomials.

Abstract

We consider the generating function $\Phi^{(N)}$ for the reciprocals $N$-th power of factorials. We show a connection of product formulas for such series with the periods for certain families of algebraic hypersurfaces. For these families we describe their singular loci. We show that these singular loci are given by zeros of the Buchstaber-Rees polynomials, which define $N$-valued group laws. We describe a generalized Frobenius method and use it to obtain special expansions for multiplication kernels in the sense of Kontsevich. Using these expansions we provide some experimental results that connect $N$-Bessel kernels and the hierarchies of the palindromic unimodal polynomials. We study the properties of such polynomials and conjecture positivity of their roots. We also discuss the connection with Kloosterman motives as a version of the mirror duality.