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

SIGMA 9 (2013), 050, 13 pages      arXiv:1105.5770      https://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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References

1. Birkhoff G.D., The generalized Riemann problem for linear differential equations and the allied problems for linear difference and $q$-difference equations, Proc. Amer. Acad. Arts Sci. 49 (1913), 521-568.
2. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
3. Gauss C.F., Disquisitiones generales circa seriem infinitam ..., in Werke, Bd. 3, Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1866, 123-162.
4. Morita T., A connection formula of the Hahn-Exton $q$-Bessel function, SIGMA 7 (2011), 115, 11 pages, arXiv:1105.1998.
5. Ohyama Y., A unified approach to $q$-special functions of the Laplace type, arXiv:1103.5232.
6. Ramis J.P., Sauloy J., Zhang C., Local analytic classification of $q$-difference equations, arXiv:0903.0853.
7. Watson G.N., The continuation of functions defined by generalized hypergeometric series, Trans. Camb. Phil. Soc. 21 (1910), 281-299.
8. Zhang C., Remarks on some basic hypergeometric series, in Theory and Applications of Special Functions, Dev. Math., Vol. 13, Springer, New York, 2005, 479-491.
9. Zhang C., Sur les fonctions $q$-Bessel de Jackson, J. Approx. Theory 122 (2003), 208-223.
10. Zhang C., Une sommation discrète pour des équations aux $q$-différences linéaires et à coefficients analytiques: théorie générale et exemples, in Differential Equations and the Stokes Phenomenon, World Sci. Publ., River Edge, NJ, 2002, 309-329.