## About Jackson $$q$$-Bessel functions. (Sur les fonctions $$q$$-Bessel de Jackson.)(French)Zbl 1023.33012

The author applies a basic Borel Laplace transformation to solve the $$q$$-difference equations $\Biggl\{ \sigma_q- \Bigl( q^{\frac\nu 2}+ q^{\frac{-\nu}{2}}\Bigr) \sigma_p+ \biggl(1+ \frac{x^2}{4}\biggr)\Biggr\} y(x)=0; \quad p= \sqrt{q};\;\sigma_qf(x)= f(qx),$ satisfied by Jackson $$q$$-Bessel functions $J_\nu^{(1)}(x;q)= \frac{(q^{\nu+1};q)_\infty} {(q;q)_\infty} \biggl( \frac x2\biggr)^\nu {}_2\varphi_1 \biggl(0,0; q^{\nu+1}; q, -\frac{x^2}{4} \biggr).$ He also obtains the connection matrices between solutions at the origin and solutions at infinity.

### MSC:

 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$
Full Text:

### References:

 [1] Adams, C.R., Linear q-difference equations, Bull. amer. math. soc., 37, 361-382, (1931) · Zbl 0002.19103 [2] Askey, R., The q-gamma and q-beta functions, Appl. anal., 8, 125-141, (1978) · Zbl 0398.33001 [3] 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, 521-568, (1913) · JFM 44.0391.03 [4] Chen, Y.; Ismail, M.E.H.; Muttalib, K.A., Asymptotics of basic Bessel functions and q-Laguerre polynomials, J. comput. appl. math., 54, 263-272, (1994) · Zbl 0829.33012 [5] Etingof, P.I., Galois groups and connection matrices of q-difference equations, Electron. res. announce. amer. math. soc., 1, 1-9, (1995) [6] Gasper, G.; Rahman, M., Basic hypergeometric series, encyclopedia of mathematics and its applications, (1990), Cambridge University Press Cambridge [7] Ismail, M.E.H., The zeros of basic Bessel functions, the functions J(nu+ax)(x), and associated orthogonal polynomials, J. math. anal. appl., 86, 1-19, (1982) · Zbl 0483.33004 [8] Lang, S., Real and functional analysis, (1993), Springer New York · Zbl 0831.46001 [9] Littlewood, J.E., On the asymptotic approximation to integral functions of zero order, Proc. London math. soc. ser., 2, 5, 361-410, (1907) · JFM 38.0450.01 [10] M. van der Put, M. Singer, Galois Theory of Difference Equations, Lecture Notes in Mathematics, Vol. 1666, Springer, Berlin, 1997. · Zbl 0930.12006 [11] Ramis, J.P.; Martinet, J., Théorie de Galois différentielle et resommation, (), 117-214 [12] J. Sauloy, Théorie de Galois des équations aux q-différences fuchsiennes, Thèse de Toulouse, 1999. [13] G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library Edition, Cambridge University Press, Cambridge, 1995. [14] Zhang, C., Développements asymptotiques q-Gevrey et séries gq-sommables, Ann. inst. Fourier, 49, 227-261, (1999) · Zbl 0974.39009 [15] Zhang, C., Transformations de q-borel – laplace au moyen de la fonction thêta de Jacobi, C. R. acad. sci. Paris Sér. I, t., 331, 31-34, (2000) · Zbl 1101.33307 [16] C. Zhang, Une sommation discrèt pour des équations aux q-différences linéaires et à coefficients analytiques: théorie générale et exemples, in: J.L.J. Braaksma, G.K. Immink, M. van der Put, J. Top (Eds.), Differential Equations and Stokes Phenomenon, World Scientific, 2002.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.