Algebraic independence of subseries over arithmetical progressions of certain exponential power series
Let \[ f(z)=\sum_{n\geq0}a_nz^n\in\mathbb{C}z, \] let \(q\in\mathbb{Z}_{>0}\) and define, for \(r=0,\ldots,q-1\), the \textit{subseries} \[ f_{q,r}(z)=\sum_{k\geq0}a_{qk+r}z^{qk+r} \] \textit{modulo} \(q\) of \(f(z)\). The authors study the algebraic independence over \(\mathbb{Q}\) of the values \(f_{q,0}(\alpha),\dots ,f_{q,q-1}(\alpha)\) at nonzero points \(\alpha\). Their main result concerns the subseries mod \(q\) of the particular entire function \((\ast)\!\!: c_1e^{\rho_1z}+c_2e^{\rho_2z}\) with nonzero algebraic \(c_1,c_2,\rho_1,\rho_2\) and reads as follows. If \(\rho_1/\rho_2\notin\mathbb{Q}(e^{2\pi i/q})\) and if \(\alpha\neq0\) is algebraic, then the field \[ \mathbb{Q}\Big(\sum_{\substack{ n\geq0,\\ n\equiv r \pmod q}}(c_1\rho_1^n+c_2\rho_2^n)\alpha^n/n! \big|\; r=0,\ldots,q-1\Big) \] has transcendence degree \(\mathrm{min}(2\varphi(q),q)\) over \(\mathbb{Q}\), where \(\varphi\) denotes Euler's totient function. Specializing the function \((\ast)\) to the exponential generating function \(\sum_{n\geq0}F_nz^n/n!\) of the Fibonacci sequence \((F_n)_{n\geq0}\), one obtains the algebraic independence over \(\mathbb{Q}\) of all numbers \[ \sum_{k\geq0}F_{qk+r}/(qk+r)! \;(r=0,\ldots,q-1) \] if \(q\) is a prime \(\neq5\). Finally subseries mod \(q\) of the function \(\sum_{n\geq0}(\sin n\theta)z^n/n!\) (or with \(\cos\) instead of \(\sin\)) are considered for \(\theta\in\pi\mathbb{Q}\). The authors' proofs appeal to Lindemann-Weierstrass theorem by using an appropriate kind of algebraic independence criterion.
scientific article; zbMATH DE number 6529303