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

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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.