Transcendental values of Bessel-Carlitz functions
Let \(\mathbb{F}_q[T]=A\) denote the ring of polynomials in one variable with coefficients in the finite field \(\mathbb{F}_q\) of prime characteristic \(p\), let \(k=\mathbb{F}_q(T)\) be its field of fractions and \(k_\infty= \mathbb{F}_q((1/T))\) the completion of \(k\) with respect to the \((1/T)\)-adic valuation \(v\) that can be continued to an algebraic closure \(\overline k\) (resp. \(\overline k_\infty)\) of \(k\) (resp. \(k_\infty)\) and to the completion \(C\) of \(\overline k_\infty\). For \(\alpha\in C\) we set \(|\alpha| =q^{d (\alpha)} =q^{-v(\alpha)}\) and \(d(0)= -\infty\). The aim of this paper is to prove convenient conditions for the transcendence over \(k\) for the value sof the Bessel-Carlitz functions which are integral series with coefficients it \(k\). For each integer \(m>0\), the Bessel function of order \(m\), as defined by Carlitz, associates with each \(z\in C\) the sum of the following series \[ J_m(z)=\sum^\infty_{k=0}\frac{(-1)^k z^{q^{n +k}}} {D_{m+k} D_k^{q^n}}. \] Define the operator \(\Delta\) by \(\Delta f(z)=f(Tz)-Tf(z)\). Theorem 1: Given \(\alpha\in C\), algebraic over \(k\) and \(\alpha\neq 0\), then for all natural integers in \(J_m(\alpha)\) and \(\Delta J_m(\alpha)\) are algebraically independent over \(k\).
scientific article; zbMATH DE number 1185573