We consider the generating function \(\Phi ^{(N)}\) for the reciprocals of the N-th power of factorials. We show a connection between product formulas for such series and the periods of certain algebraic hypersurface families. For these families, we describe their singular loci. We show that these singular loci are given by the 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 palindromic unimodal polynomials. We study the properties of such polynomials and conjecture the positivity of their roots. We also discuss the connection with Kloosterman motives as a version of mirror duality.
