Jackson–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.

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

They can be reduced to the Bessel function by the continuous limit:

There is a connection formula between the first and second Jackson q-Bessel function (Gasper & Rahman (2004)):

For integer order, the q-Bessel functions satisfy

By using the relations (Gasper & Rahman (2004)):

we obtain

Hahn mentioned that ${\displaystyle J_{\nu }^{(2)}(x;q)}$ has infinitely many real zeros (Hahn (1949)). Ismail proved that for ${\displaystyle \nu >-1}$ all non-zero roots of ${\displaystyle J_{\nu }^{(2)}(x;q)}$ are real (Ismail (1982)).

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