Geodesic
On estimates, unimprovable with respect to height, of some linear forms
Sbornik. Mathematics, Tome 52 (1985) no. 2, pp. 407-421 
A. I. Galochkin. On estimates, unimprovable with respect to height, of some linear forms. Sbornik. Mathematics, Tome 52 (1985) no. 2, pp. 407-421. http://geodesic.mathdoc.fr/item/SM_1985_52_2_a6/
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     title = {On estimates, unimprovable with respect to height, of some linear forms},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Shidlovskii A. B., “Ob arifmeticheskikh svoistvakh znachenii analiticheskikh funktsii”, Trudy MIAN, 132 (1973), 169–202 | Zbl

[2] Osgood C., “Some theorems on diophantine approximation”, Trans. Amer. Math. Soc., 123:1 (1966), 64–87 | DOI | MR | Zbl

[3] Galochkin A. I., “Otsenki snizu lineinykh form ot znachenii nekotorykh gipergeometricheskikh funktsii”, Matem. zametki, 8:1 (1970), 19–28 | Zbl

[4] Galochkin A. I., “Utochnenie otsenok nekotorykh lineinykh form”, Matem. zametki, 20:1 (1976), 35–45 | Zbl

[5] Korobov A. N., “Otsenki nekotorykh lineinykh form”, Vestn. MGU. Seriya matem. mekh., 1981, no. 6, 36–40 | MR | Zbl

[6] Uitteker E. T., Vatson D. Ya., Kurs sovremennogo analiza, ch. II, Fizmatgiz, M., 1963

[7] Prakhar K., Raspredelenie prostykh chisel, Mir, M., 1967 | MR

[8] Galochkin A. I., “O poluchenii tochnykh po vysote otsenok nekotorykh lineinykh form”, Teoriya transtsendentnykh chisel i ee prilozheniya, Tezisy dokladov Vsesoyuznoi konf., Izd-vo MGU, M., 1983, 21–22

[9] Galočkin A. I., “Sur des estimations précises par rapport à la hauteur de certaines formes linéaires”, Proc. conference: Approximations diophantiennes et nombres transcendants. Luminy, 1982, Progress in Math., Birkhäuser, 1983