Product formulas for the Higher Bessel functions

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.