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Hahn–Exton q-Bessel function

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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)):

J ν + 1 ( 3 ) ( x ; q ) = ( 1 − q ν x + x ) J ν ( 3 ) ( x ; q ) − J ν − 1 ( 3 ) ( x ; q ) . {\displaystyle J_{\nu +1}^{(3)}(x;q)=\left({\frac {1-q^{\nu }}{x}}+x\right)J_{\nu }^{(3)}(x;q)-J_{\nu -1}^{(3)}(x;q).}

Alternative Representations

Integral Representation

The Hahn–Exton q-Bessel function has the following integral representation (see Ismail and Zhang (2018)):

J ν ( 3 ) ( z ; q ) = z ν π log ⁡ q − 2 ∫ − ∞ ∞ exp ⁡ ( x 2 log ⁡ q 2 ) ( q , − q ν + 1 / 2 e i x , − q 1 / 2 z 2 e i x ; q ) ∞ d x . {\displaystyle J_{\nu }^{(3)}(z;q)={\frac {z^{\nu }}{\sqrt {\pi \log q^{-2}}}}\int _{-\infty }^{\infty }{\frac {\exp \left({\frac {x^{2}}{\log q^{2}}}\right)}{(q,-q^{\nu +1/2}e^{ix},-q^{1/2}z^{2}e^{ix};q)_{\infty }}}\,dx.}
( a 1 , a 2 , ⋯ , a n ; q ) ∞ := ( a 1 ; q ) ∞ ( a 2 ; q ) ∞ ⋯ ( a n ; q ) ∞ . {\displaystyle (a_{1},a_{2},\cdots ,a_{n};q)_{\infty }:=(a_{1};q)_{\infty }(a_{2};q)_{\infty }\cdots (a_{n};q)_{\infty }.}

Hypergeometric Representation

The Hahn–Exton q-Bessel function has the following hypergeometric representation (see Daalhuis (1994)):

J ν ( 3 ) ( x ; q ) = x ν ( x 2 q ; q ) ∞ ( q ; q ) ∞   1 ϕ 1 ( 0 ; x 2 q ; q , q ν + 1 ) . {\displaystyle J_{\nu }^{(3)}(x;q)=x^{\nu }{\frac {(x^{2}q;q)_{\infty }}{(q;q)_{\infty }}}\ _{1}\phi _{1}(0;x^{2}q;q,q^{\nu +1}).}

This converges fast at x → ∞ {\displaystyle x\to \infty } . It is also an asymptotic expansion for ν → ∞ {\displaystyle \nu \to \infty } .

References

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