# Jackson q-Bessel function

In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by Jackson (1903, 1903b, 1905, 1905b). The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function.

## Definition

The three Jackson q-Bessel functions are given in terms of the Pochhammer symbol and the basic hypergeometric function φ by

${\displaystyle J_{\nu }^{(1)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{2}\phi _{1}(0,0;q^{\nu +1};q,-x^{2}/4)}$
${\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{0}\phi _{1}(;q^{\nu +1};q,-x^{2}q^{\nu +1}/4)}$
${\displaystyle J_{\nu }^{(3)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{1}\phi _{1}(0;q^{\nu +1};q,qx^{2}/4)}$

## References

• Ismail, Mourad E. H. (1982), "The zeros of basic Bessel functions, the functions Jν +ax(x), and associated orthogonal polynomials", Journal of Mathematical Analysis and Applications, 86 (1): 1–19, doi:10.1016/0022-247X(82)90248-7, ISSN 0022-247X, MR 649849
• Jackson, F. H. (1903), "On generalized functions of Legendre and Bessel", Transactions of the Royal Society of Edinburgh, 41: 1–28
• Jackson, F. H. (1903), "Theorems relating to a generalization of the Bessel functions", Transactions of the Royal Society of Edinburgh, 41: 105–118
• Jackson, F. H. (1904), "Theorems relating to a generalization of Bessel's function.", Transactions of the Royal Society of Edinburgh, 41: 399–408, doi:10.1017/s0080456800034475, JFM 36.0513.02
• Jackson, F. H. (1905), "The Application of Basic Numbers to Bessel's and Legendre's Functions", Proceedings of the London Mathematical Society, 2 (1): 192–220, doi:10.1112/plms/s2-2.1.192
• Jackson, F. H. (1905), "The Application of Basic Numbers to Bessel's and Legendre's Functions (Second paper)", Proceedings of the London Mathematical Society, 3 (1): 1–23, doi:10.1112/plms/s2-3.1.1