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JOURNALS // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika // Archive

Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1982 Number 1, Pages 11–14 (Mi vmumm4127)

Mathematics

The poles of Padé approximants to $_1F_1(1;c;z)$

D. V. Pannikov


Abstract: We construct some regions without zeros of the confluent hypergeometric function $_1F_1(-n;d;z)$ ($n\in\mathbf N, d\in\mathbf C$). The main result is as follows. If
$$ -n+\frac7{16}\geq\operatorname{Re}(d),\quad_1F_1(-n;d;z)=0, $$
then
$$ -\operatorname{Re}(d)-\operatorname{Im}(d)\operatorname{tg}\biggl(\frac{\arg(z)}2\biggr) \geq|z|>\operatorname{Re}(z)+2\biggl(-n+\frac7{16}- \operatorname{Re}(d)\biggr). $$


UDC: 517.53

Received: 06.03.1981



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