Numerically satisfactory solutions of Kummer recurrence relations

Springer Science and Business Media LLC - Tập 111 - Trang 109-119 - 2008
Javier Segura1, Nico M. Temme2
1Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Santander, Spain
2CWI Amsterdam, The Netherlands

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.

Tài liệu tham khảo

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