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 (1906a, 1906b, 1905a, 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 q-Pochhammer symbol and the basic hypergeometric function ϕ {\displaystyle \phi } by
J ν ( 1 ) ( x ; q ) = ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ ( x / 2 ) ν
2 ϕ 1 ( 0 , 0 ; q ν + 1 ; q , − x 2 / 4 ) , | x | < 2 , {\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),\quad |x|<2,}
J ν ( 2 ) ( x ; q ) = ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ ( x / 2 ) ν
0 ϕ 1 ( ; q ν + 1 ; q , − x 2 q ν + 1 / 4 ) , x ∈ C , {\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),\quad x\in \mathbb {C} ,}
J ν ( 3 ) ( x ; q ) = ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ ( x / 2 ) ν
1 ϕ 1 ( 0 ; q ν + 1 ; q , q x 2 / 4 ) , x ∈ C . {\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),\quad x\in \mathbb {C} .}
They can be reduced to the Bessel function by the continuous limit:
lim q → 1 J ν ( k ) ( x ( 1 − q ) ; q ) = J ν ( x ) , k = 1 , 2 , 3. {\displaystyle \lim _{q\to 1}J_{\nu }^{(k)}(x(1-q);q)=J_{\nu }(x),\ k=1,2,3.}
There is a connection formula between the first and second Jackson q-Bessel function (Gasper & Rahman (2004)):
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