Electron. J. Differential Equations, Vol. 2017 (2017), No. 31, pp. 1-20.

A q-analogue of Kummer's equation

Lukun Jia, Jinfa Cheng, Zhaosheng Feng

Abstract:
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 ${}_1\Phi_1$.

Submitted May 9, 2016. Published January 29, 2017.
Math Subject Classifications: 39A13, 39A05, 33D15.
Key Words: q-analogue, Kummer's equation; Frobenius method; contiguous relations.

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Lukun Jia
School of Mathematical Science
Xiamen University
Xiamen, Fujian 361005, China
email: jialukun2005@163.com
Jinfa Cheng
School of Mathematical Science
Xiamen University
Xiamen, Fujian 361005, China
email: jfcheng@xmu.edu.cn
Zhaosheng Feng
School of Mathematical and Statistical Sciences
University of Texas-Rio Grande Valley
Edinburg, Texas 78539, USA
email: zhaosheng.feng@utrgv.edu

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