In mathematics, the Hahn–Exton q-Bessel function or the third Jackson q-Bessel function is a q-analog of the Bessel function, and satisfies the Hahn-Exton q-difference equation (Swarttouw (1992)). This function was introduced by Hahn (1953) in a special case and by Exton (1983) in general. The Hahn–Exton q-Bessel function is given by
J ν ( 3 ) ( x ; q ) = x ν ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ ∑ k ≥ 0 ( − 1 ) k q k ( k + 1 ) / 2 x 2 k ( q ν + 1 ; q ) k ( q ; q ) k = ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ x ν
1 ϕ 1 ( 0 ; q ν + 1 ; q , q x 2 ) . {\displaystyle J_{\nu }^{(3)}(x;q)={\frac {x^{\nu }(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}\sum _{k\geq 0}{\frac {(-1)^{k}q^{k(k+1)/2}x^{2k}}{(q^{\nu +1};q)_{k}(q;q)_{k}}}={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}x^{\nu }{}_{1}\phi _{1}(0;q^{\nu +1};q,qx^{2}).}
ϕ {\displaystyle \phi } is the basic hypergeometric function.
Properties
Zeros Koelink and Swarttouw proved that J ν ( 3 ) ( x ; q ) {\displaystyle J_{\nu }^{(3)}(x;q)} has infinite number of real zeros. They also proved that for ν > − 1 {\displaystyle \nu >-1} all non-zero roots of J ν ( 3 ) ( x ; q ) {\displaystyle J_{\nu }^{(3)}(x;q)} are real (Koelink and Swarttouw (1994)). For more details, see Abreu, Bustoz & Cardoso (2003). Zeros of the Hahn-Exton q-Bessel function appear in a discrete analog of Daniel Bernoulli's problem about free vibrations of a lump loaded chain (Hahn (1953), Exton (1983))
Derivatives For the (usual) derivative and q-derivative of J ν ( 3 ) ( x ; q ) {\displaystyle J_{\nu }^{(3)}(x;q)} , see Koelink and Swarttouw (1994). The symmetric q-derivative of J ν ( 3 ) ( x ; q ) {\displaystyle J_{\nu }^{(3)}(x;q)} is described on Cardoso (2016).
Recurrence Relation The Hahn–Exton q-Bessel function has the following recurrence relation (see Swarttouw (1992)):
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