Lower estimates of linear forms in the values of the Kummer function with an irrational parameter (Q805662)

The author considers arithmetic properties of the Kummer function of type \[ \psi (z)=\sum^{\infty}_{\nu =0}z^{\nu}\prod^{\nu}_{x=1}\frac{x+\alpha}{x(x+\beta)b(x)}, \] where \(b(x)=(x+\beta_ 1)...(x+\beta_ u)\), \(u\geq 0\) \((u=0\) means \(b(x)=1)\); \(\alpha\in {\mathbb{Q}}\setminus {\mathbb{Z}}\), \(\beta\in I\setminus {\mathbb{Q}}\) (I denotes an imaginary quadratic field); \(\beta_ 1,...,\beta_ u\in {\mathbb{Q}}\), \(\alpha -\beta_ j\not\in {\mathbb{Z}}\), \(\beta_ j\neq -1,-2,-3,..\). \((j=1,...,u)\). He proves that for \(\xi\in I\setminus \{0\}\) and any \(\epsilon >0\) \[ | \sum^{m}_{j=1}h_ j\psi^{(j-1)}(\xi)| \quad >\quad H^{1-2m-\epsilon}\quad (m=2+u), \] provided \(H=\max (| h_ 1|,...,| h_ m|)>H_ 0(\alpha,\beta,\beta_ j,\xi,\epsilon,I)\), where \((h_ 1,...,h_ m)\) is any nontrivial tuple of integers of I.