Hypergeometric functions for function fields
type: Item
instance of: scholarly article
title: Hypergeometric functions for function fields
zbMATH Open document ID: 0838.11043
author: Dinesh S. Thakur
published in: Finite Fields and their Applications
publication date: 1996-06-03T00:00:00Z
review text: Let \(\{a,b,c\}\) be complex constants. Then the famous Gauss hypergeometric equation is given by \[ z(1 - z) {d^2y \over dz^2} + \bigl( c - (a + b + 1) z \bigr) {dy \over dz} - aby = 0. \] One defines the Pochhammer symbol \((a)_n\) by \((a)_0 : = 1\) and for \(n > 1\), \((a)_n : = a(a + 1) (a + 2) (a + 3) \cdots (a + n - 1)\). A series solution to the Gauss differential equation is given by \(F(a,b;c;z) : = \sum^\infty_{n = 0} {(a)_n (b)_n \over n! (c)_n} z^n\). (So one has the identity \((1 - z)^s = F(- s,b;b;z).)\) The hypergometric functions \(F(a,b;c;z)\) have many applications to mathematics and physics. As an example, there is the beautiful application by Dwork of the above equation to \(p\)-adic unit roots of characteristic polynomials of Frobenius [see \textit{N. M. Katz}, Sémin. Bourbaki 1971/72, Exp. No. 409, 167-200 (1973; Zbl 0259.14007)]. Now let \({\mathbf A} = \mathbb{F}_q [T]\), \(q = p^{m_0}\), \({\mathbf k} = \mathbb{F}_q (T)\) and \({\mathbf K} = \mathbb{F}_q ((1/T))\). Let \(D_0 : = 1\) and for \(i > 0\) \(D_i : = (T^{q^i} - T) D^q_{i - 1}\). These \(\{D_i \}\) can be fit into the Carlitz exponential \(e_C (x) = \sum^\infty_{i = 0} z^{q^i}/D_i\). This power series is entire and, since we are in characteristic \(p\), we find that \(e_C (x)\) is also \(\mathbb{F}_q\)-linear --- the zeros \(\Lambda\) of \(e_C (x)\) are thus an \(\mathbb{F}_q\)-vector space; one can see that there is an element \(\xi \neq 0\) with \(\Lambda = {\mathbf A} \xi\). Moreover, \(e_C (x)\) satisfies the well-known Carlitz functional equation \(e_C (Tx) = Te_C (x) + e^q_C (x)\). Thus \(e_C (T^2x) = T^2 e_C (x) + (T^q + T) e^q_C (x) + e_C^{q^2} (x)\) and so on. Just as the classical functional equation \(e^{nz} = (e^z)^n\) leads to the usual action of \(\mathbb{Z}\) on \(\mathbb{C}^*\), the Carlitz functional equation leads to a new action of \({\mathbf A}\) on the additive group called the Carlitz module. In classical theory, the cyclotomic extensions of \(\mathbb{Q}\) are…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 33C99
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 759428
zbMATH Keywords: arithmetic hypergeometric functions
zbMATH Keywords: Carlitz exponential
zbMATH Keywords: Carlitz functional equation
zbMATH Keywords: geometric extensions
zbMATH Keywords: geometric hypergeometric functions
zbMATH Keywords: Pochhammer symbols
reviewed by: David Goss
MaRDI profile type: Publication
full work available at URL: ffta.1995.1017
OpenAlex ID: W2060193645
DOI: 10.1006/FFTA.1995.1017
description: scientific article; zbMATH DE number 759428