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Numerically satisfactory solutions of Kummer recurrence relations

Authors:	Javier Segura Nico M Temme

Institution:	(1) Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005 Santander, Spain;(2) CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

Abstract:	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.

Keywords:	Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 33C15 39A11 41A60 65D20
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