Numerically satisfactory solutions of Kummer recurrence relations
Tóm tắt
Pairs of numerically satisfactory solutions as
$${n \rightarrow \infty}$$
for the three-term recurrence relations satisfied by the families of functions
$${_1{\rm F}_1(a+\epsilon_1 n; b +\epsilon_2 n; z)}$$
,
$${\epsilon_i \in {\mathbb Z}}$$
, are given. It is proved that minimal solutions always exist, except when
$${\epsilon_2=0}$$
and z is in the positive or negative real axis, and that
$${_1{\rm F}_1 (a+ \epsilon_1 n; b +\epsilon_2 n; z)}$$
is minimal as
$${n \rightarrow + \infty}$$
whenever
$${\epsilon_2 > 0}$$
. The minimal solution is identified for any recurrence direction, that is, for any integer values of
$${\epsilon_1}$$
and
$${\epsilon_2}$$
. When
$${\epsilon_2 \neq 0}$$
the confluent limit
$${\lim_{b \rightarrow \infty}\,_1{\rm F}_1(\gamma b; b; z)= e^{\gamma z}}$$
, with
$${\gamma \in {\mathbb C}}$$
fixed, is the main tool for identifying minimal solutions together with a connection formula; for
$${\epsilon_2=0}$$
,
$${\lim_{a \rightarrow +\infty}\,_1{\rm F}_1(a; b; z)/_0{\rm F}_1(; b; az)=e^{z/2}}$$
is the main tool to be considered.
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