Explicit formulas for the exponential and logarithm of the Carlitz-Tate twist, and applications (Q6583682)

The main purpose of this paper is the establishment of explicit formulas for the exponential and logarithm of the higher dimension Carlitz module (the \(n\)-th tensor power) \(C^{\otimes n}:A \to M_n({\mathbb C}_{\infty}\{\tau\})\) given by\N\begin{multline*}\NC^{\otimes n}(\theta)=(\theta I_n +N) \tau^0 +E\tau\\\N=\left(\left[ \begin{array}{ccccc} \theta&0&0&\cdots&0\\\N0&\theta&0&\cdots&0\\\N\vdots&\vdots&\vdots&\ddots&\vdots\\\N0&0&0&\cdots&0\\\N0&0&0&\cdots&\theta \end{array} \right]+ \left[ \begin{array}{ccccc} 0&1&0&\cdots&0\\\N0&0&1&\cdots&0\\\N\vdots&\vdots&\vdots&\ddots&\vdots\\\N0&0&0&\cdots&1\\\N0&0&0&\cdots&0 \end{array} \right] \right)\tau^0+ \left[ \begin{array}{ccccc} 0&0&0&\cdots&0\\\N0&0&0&\cdots&0\\\N\vdots&\vdots&\vdots&\ddots&\vdots\\\N0&0&0&\cdots&0\\\N1&0&0&\cdots&0 \end{array} \right]\tau,\N\end{multline*}\Nwhere \(A={\mathbb F}_q[\theta]\). The exponential and the logarithm are defined by \(\exp_n:=I_n\tau^0+\sum_{i\geq 1}E(i)\tau^i\) and \(\log_n:= I_n\tau^0+\sum_{i\geq 1}L(i) \tau^i\), where \(E(i)\) and \(L(i)\) are defined recursively.\N\NThe main results are given in Section 2 (Theorems 2.3, 2.6 and Corollaries 2.4, 2.7). These results give the following recursion formulas:\N\begin{gather*}\ND_i^n\cdot E(i)_{l,m}=\sum_{j=0}^{m-1} \binom {l-1}j [i]^{(l-1)-j} \sum_{s=1}^{m-1-j} (-1)\binom ns \sum_{t=1}^{i}\left(\frac{D_t^n \cdot E(t)_{1,m-j-s}}{[t]^s}\right)^{q^{i-t}},\\\NL_i^n\cdot L(i)_{l,m}=\sum_{j=0}^{n-l}\binom{n-m}{j}(-[i])^{(n-m)-j} \sum_{s=1}^{n-l-j}(-1)\binom ns \sum_{t=1}^i \frac{L_t^n\cdot L(t)_{l+j+s,n}}{(-[t])^s},\N\end{gather*}\Nwith the notations given at the beginning of the paper.\N\NIn the last half of this paper, transcendence problems are investigated for entries of \(\exp_n\) and \(\log_n\) by means of hypergeometric functions.