On estimates, unimprovable with respect to height, of some linear forms

A. I. Galochkin

Sbornik. Mathematics, Tome 52 (1985) no. 2, pp. 407-421

Résumé

Lower and upper bounds that differ from each other only by a constant factor are obtained for linear forms in values of the function $$ \psi(z)=\sum_{\nu=0}^\infty\frac{z^\nu}{b^{(s+1)\nu}\nu!\,[\lambda_1+1,\nu]\dots[\lambda_s+1,\nu]}, $$ $[\lambda+1,\nu]=(\lambda+1)\dots(\lambda+\nu)$, $[\lambda+1,0]=1$ and its $s$ successive derivatives at the point $z=\frac1b$ under the condition that $a,b$ and $a\lambda_1,\dots,a\lambda_s$ are integers in some imaginary quadratic field. Bibliography: 9 titles.