Arithmetic properties of some \(p\)-adic numbers (Q1596182)

Let \(\Omega_p\) denote a supplement of the algebraic closure \(\overline{\mathbb Q_p}\) of the field \(\mathbb Q_p\). The author discusses problems of transcendence and linear independence of the elements from \(\Omega_p\) over \(\mathbb Q_p\). Especially the author considers a generalized hypergeometric series \[ F(a_1,\cdots ,a_m,z)=\sum\limits_{n=0}^\infty (a_1)_n\cdots (a_m)_nz^n/n!, \] where \(a_1,\cdots ,a_m\) are nonzero rational numbers different from negative integers, \((a)_0=1\), \((a)_n=a(a+1)\cdots (a+n-1)\), \(n\geq 1\). Denote \(f_0(z)=F(a_1,\cdots ,a_m,z)\), \(f_1(z)=F(a_1+1,a_2,\cdots ,a_m,z),\break \cdots,f_{m-1}(z)=F(a_1+1,\cdots ,a_{m-1}+1,a_m,z)\). He shows that the numbers \[ f_0(p^{1/n}),\;f_1(p^{1/n}),\cdots,f_{m-1}(p^{1/n}) \] are linearly independent over \(\mathbb Q_p\). Let \(S_p(n)\) be the sum of digits in the \(p\)-ary decomposition of a number \(n\). For any positive rationals \(r,s,t\) set \[ \gamma (n)=\frac{r}{n-S_p(n)}+s+tn. \] It is proved that the series \(\sum_{n=1}^\infty (n!)^{\gamma (n)}\) converges in \(\mathbb C_p\) to an element algebraic over \(\mathbb Q_p\). At the same time, the series converges in \(\mathbb C_q\) for any \(q\neq p\) and the limit is transcendental over \(\mathbb Q_q\).