SIGMA 7 (2011), 115, 11 pages      arXiv:1105.1998      https://doi.org/10.3842/SIGMA.2011.115

A Connection Formula of the Hahn-Exton q-Bessel Function

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

Received May 11, 2011, in final form December 14, 2011; Published online December 16, 2011

Abstract
We show a connection formula of the Hahn-Exton q-Bessel function around the origin and the infinity. We introduce the q-Borel transformation and the q-Laplace transformation following C. Zhang to obtain the connection formula. We consider the limit p→1⁻ of the connection formula.

Key words: Hahn-Exton q-Bessel function; q-Borel transformation; connection problems.

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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. 49 (1913), 521-568.
2. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004.
3. Hahn W., Beiträge zur Theorie der Heineschen Reihen. Die 24 Integrale der Hypergeometrischen q-Differenzengleichung. Das q-Analogon der Laplace-Transformation, Math. Nachr. 2 (1949), 340-379.
4. Olver F.W.J., Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press, New York - London, 1974.
5. Swarttouw R.F., Meijer H.G., A q-analogue of the Wronskian and a second solution of the Hahn-Exton q-Bessel difference equation, Proc. Amer. Math. Soc. 129 (1994), 855-864.
6. Watson G.N., The continuation of functions defined by generalized hypergeometric series, Trans. Camb. Phil. Soc. 21 (1910), 281-299.
7. 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.
8. Zhang C., Sur les fonctions q-Bessel de Jackson, J. Approx. Theory 122 (2003), 208-223.
9. Zhang C., Une sommation discrè pour des équations aux q-différences linéaires et à coefficients, analytiques: théorie générale et exemples, in Differential Equations and Stokes Phenomenon, World Sci. Publ., River Edge, NJ, 2002, 309-329.