Pairs of numerically satisfactory solutions as $n\rightarrow \infty$ for the three-term recurrence relations satisfied by the families of functions $_1\mbox{F}_1(a+\epsilon_1 n; b +\epsilon_2 n;z)$, $\epsilon_i \in {\mathbb Z}$, 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 $_1\mbox{F}_1 (a+\epsilon_1 n; b +\epsilon_2 n;z)$ 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 $\lim_{b\rightarrow \infty}{}_1\mbox{F}_1(\gamma b;b;z)=e^{\gamma z}$, with $\gamma\in{\mathbb C}$ fixed, is the main tool for identifying minimal solutions together with a connection formula; for $\epsilon_2=0$, $\lim_{a\rightarrow +\infty} {}_1\mbox{F}_1(a;b;z) /{}_0\mbox{F}_1(;b;az)=e^{z/2}$ is the main tool to be considered.

Additional Metadata
MSC Computation of special functions, construction of tables (msc 65D20)
Publisher Springer
Journal Numerische Mathematik
Citation
Segura, J, & Temme, N.M. (2008). Numerically satisfactory solutions of Kummer recurrence relations. Numerische Mathematik, 111(1), 109–119.