Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 050, 13 pages      arXiv:1105.5770      http://dx.doi.org/10.3842/SIGMA.2013.050

A Connection Formula for the $q$-Confluent Hypergeometric Function

Takeshi Morita
Graduate School of Information Science and Technology, Osaka University, 1-1 Machikaneyama-machi, Toyonaka, 560-0043, Japan

Received October 09, 2012, in final form July 21, 2013; Published online July 26, 2013

Abstract
We show a connection formula for the $q$-confluent hypergeometric functions ${}_2\varphi_1(a,b;0;q,x)$. Combining our connection formula with Zhang's connection formula for ${}_2\varphi_0(a,b;-;q,x)$, we obtain the connection formula for the $q$-confluent hypergeometric equation in the matrix form. Also we obtain the connection formula of Kummer's confluent hypergeometric functions by taking the limit $q\to 1^{-}$ of our connection formula.

Key words: $q$-Borel-Laplace transformation; $q$-difference equation; connection problem; $q$-confluent hypergeometric function.

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