A q-analogue of Kummer's equation

• Main Authors: Lukun Jia, Jinfa Cheng, Zhaosheng Feng
• Format: Article
• Language: English
• Published: Texas State University 2017-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: q-analogue, Kummer's equation; Frobenius method; contiguous relations
• Online Access: http://ejde.math.txstate.edu/Volumes/2017/31/abstr.html

In this article we define a q-analogue of Kummer's equation. It has two singular points. Near the singular point at zero, using the Frobenius method, we obtain two linearly independent series solutions in any one of three cases according to the difference of roots of the characteristic equation. Near the singular point at infinity, given that the only formal series solution is divergent, we find two integral solutions which are convergent under some condition. Finally, using the q-analogue of Kummer's equation, we deduce six contiguous relations about the q-hypergeometric series ${}_1\Phi_1$.