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.


33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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