Algebraic independence of arithmetic gamma values and Carlitz zeta values

Abstract

We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for Fq[θ] and provide complete algebraic independence results for them.

Hypergeometric functions for function fields and transcendence

“…With the help of Theorem 5, we can also give another proof of the main result of [11] (originally proved by automata method) proving transcendence of the Carlitz-Goss gamma values at non-natural p-adic integers. Note that the special case giving the transcendence of the values at fractions treated in [15], and [2] by automata method has been vastly generalized to determination of full algebraic relations between the values at fractions in [4], but the period method of [3,12,4] does not apply to non-fractional p-adic integers.…”

Hyperderivatives of periods and quasi-periods for Anderson $t$-modules

“…Other investigations by Anderson, Brownawell, Chang, Denis, Thakur, Yu, and many others have produced transcendence results on function field Γ-values [2], [14], [22], [23], [74], [76], [77]; Drinfeld logarithms and quasi-logarithms [8]- [10], [20], [21], [28], [32], [34], [62], [69], [82], [83], [85], [87]; zeta values and multiple zeta values [16]- [19], [24]- [26], [42], [44], [53], [86], [87]; and of particular interest to the present paper, hyperderivatives of Drinfeld logarithms and quasi-logarithms [10]- [13], [30]- [32], [54].…”

Geometric Gamma Values and Zeta Values in Positive Characteristic

“…One may ask about algebraic independence questions for the arithmetic Γ-function and its relations with Carlitz ζ-values as well. These questions are addressed in [8].…”