Title: Product formulas for the Higher Bessel functions Show Chinese title

Title: 高阶贝塞尔函数的乘积公式

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. Show Chinese abstract

Abstract: 我们考虑生成函数$\Phi^{(N)}$对于阶乘的倒数的$N$次幂。 我们展示了这类级数的乘积公式与某些代数超曲面族的周期之间的联系。 对于这些族，我们描述了它们的奇异点集。 我们证明这些奇异点集由Buchstaber-Rees多项式的零点给出，这些多项式定义了$N$-值的群运算。 我们描述了一种广义的Frobenius方法，并用它来获得Kontsevich意义上的乘法核的特殊展开式。 利用这些展开式，我们提供了一些实验结果，将$N$-贝塞尔核与回文单峰多项式的层次结构联系起来。 我们研究了这些多项式的性质，并猜想其根的正性。 我们还讨论了与Kloosterman动机的联系，作为镜像对偶的一种形式。