Mathematical Sciences: Hypergeometric Functions, Zeta and Gamma Values in Finite Characteristic

研究者将探索数论和几何之间的类比。超几何函数是数学和数学物理中最重要的特殊函数之一。在数学和物理的各个分支中出现的许多函数——贝塞尔、勒让德和雅可比函数，以及许多有趣的正交多项式——都是超几何函数的特殊情况。同样，q-超几何函数也出现在数学物理的研究中。研究者介绍了函数场的类似物，他将发展它们的理论和应用。数论中的一个经典问题是理解黎曼ζ函数值的本质。欧拉证明了偶数s处的值是圆周率的有理倍数。对于奇数s，唯一已知的结果是函数在s=3时是无理数。Carlitz考虑了一种类似于ζ函数的方法，即在有限域内对带系数的一元多项式求和，并建立了欧拉结果的类似物。在与Anderson的联合工作中，PI证明了对于任何正整数s，其值都是代数数的对数，因此是超越的。还证明了欧拉的结果对奇数s的类比在这种情况下不成立。提议者计划将这个结果推广到一般的函数域。ζ函数的消失顺序给出了重要的算术信息。PI计划完成他已经获得的关于这些订单的新的部分结果。此外，PI还计划参照安德森最近的研究成果，研究特殊值与切圆理论之间的关系。经典函数的特殊值也得到了广泛的研究。PI为函数场引入了一个新的函数。他建立了一般函数场在有限素数处的插值和有理函数场在无限素数处的插值的特殊值结果。他计划解决剩下的案子。本研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等各种应用领域不可或缺的工具。

Thakur 9623187 The investigator will explore analogies between Number Theory and Geometry. The hypergeometric function is one of the most important special functions in mathematics and mathematical physics. Many of the functions that arise in various branches of mathematics and physics-Bessel, Legendre, and Jacobi functions, and many interesting orthogonal polynomials-are special cases of the hypergeometric function. Similarly, the q-hypergeometric functions arise in the study of mathematical physics. The investigator has introduced an analogue for function fields, and he will develop their theory and applications. A classical problem in Number Theory is to understand the nature of the values of the Riemann zeta-function. Euler showed that the values at even integers s are rational multiples of powers of pi. For odd s, the only known result is that the zeta-function is irrational at s=3. Carlitz considered an analogue of the zeta-function by taking a sum over monic polynomials with coefficients in finite fields, and established an analogue of Euler's result. In the joint work with Anderson, the PI showed that for any positiveminteger s, the values are logarithms of algebraic numbers, and are therefore transcendental. It was also shown that the analogue of Euler's result for odd s does not hold in this case. The proposer plans to generalize thismresult to general function fields. The order of vanishing of zeta functions gives important arithmetic information. The PI plans to complete new partial results that he has obtained about these orders. Also, the PI plans to workmout the relations of special values with the cyclotomic theory, in light of recent work of Anderson. The special values of the classical gamma function have also been extensively studied. The PI introduced a new gamma function for function fields. He has established special values results for its interpolations at finite primes for general function fields and for interpolati on at the infinite prime only for the rational function field. He plans to settle the remaining case. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.